?_<ïÿÿÿÿîÉ'E < lpe¦>… …‚ …‚ÿ‚….ƒ1‚ ‡  €†&ÿƒ‰Àÿ¤ÿ@Ä “& MathType…ûþ傎PSymbol‚…-‡2 &ƒ(…ûþ傎PSymbol‚…-…ð‡2 &ƒ)…û•ý₎PSymbol…-…ð‡2 iVƒ(…û•ý₎PSymbol…-…ð‡2 iøƒ) …úƒ"…-‡J‡‚…ûþ傎PSymbol‚…-…ð‡2 T ƒ(…ûþ傎PSymbol‚…-…ð‡2 T ƒ)…û•ý₎PSymbol…-…ð‡2 û… ƒ(…û•ý₎PSymbol…-…ð‡2 û%ƒ)…û•ý₎PSymbol…-…ð‡2 i„ ƒ(…û•ý₎PSymbol…-…ð‡2 i&ƒ)‡x ‡°„û€þ‚Œ Arial…-…ð ‡2 :…x¼ ‡2 ²…tk ‡2 ´…tk ‡2 7Ý…tk ‡2 f …y¼ ‡2 à …tk ‡2 É …tk ‡2 7 …tk„û€þ‚ŽPSymbol …-…ð ‡2 …=Ó ‡2 7«…+Ó ‡2 2 …=Ó ‡2 ÉÙ…-Ó ‡2 7Ù…+Ó„û€þ‚Œ Arial…-…ð ‡2 ª…2× ‡2 7À…1× ‡2 Éï …1× ‡2 7î …1ׄû ÿ‚Œ Arial …-…ð ‡2 ‹U…2} ‡2 ‚…2} ‡2 ‹ƒ…2}„û€þ‚Œ Arial…-…ð ‡2 ¬…,k ‡2 Ú….k †& ÿ…û ‚¼"Systemn…-…ð…-…ð ‡2 ‹U¿œôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœœôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœ! k,cMc,cMk,cMc,cMk,cMc,cMk,cMc,c ½c-c,c-c-c-c,c-c-c-c,c-c-c-c,c-cœôœôœôœôœôœôœôœôœôœôœôœôœôœôœÿ½÷½÷½÷½÷½÷½÷½÷½÷½÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½œôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœÀk,cMc,cMk,cMc,cMk,cMc,cMk,cMc,cc-c-c-c,c-c-c-c,c-c-c-c,c-c-c-cAŸc,cMk,cMc,cMk,cMc,cMk,cMc,cMk,c`‰ .+#4u4X‚@¿½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½c,cMk,cMc,cMk,cMc,cMk,cMc,cMk,c!ÿœôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½ÿ‹€œ+¼ƒøÿu¿ :Dë¿ø9D¹ÿœôœôœôœôœ-c-côœôœôœôœôœôœyÎTML authoring tool bar.HTMLMa½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷ÿ½HTMLTOOLSCUSTOMIZE_ HTMLRESBAE`HTMLFILEPAGESETUPaHEADERDGHANDLERbDOKUMENTSCHLIESENc½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½°÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½œôœôœ÷½ôœ÷½†÷½÷½÷½ÿbÿmÿ oÿ6&ÿ8ÿÿ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½Š*ä@PšA+éjBVjšÀ‹ð öu½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½–õk C€à7yÃAC Ø…W4vCš™î¹?{®½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½STREAMuHTMLTOOLSMACROv NUMBEœôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœhics.(ÿ%¼ €ADFILENAME{?/ÿg/ ·>';½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½Ä÷½÷½indows. This version of Interne½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½/$lpM4@  <‚ …‚ÿ‚….ƒ1‚ ‡  †&ÿƒ‰Àÿ·ÿ`· “& MathTypeP„û€þ‚¼Œ Arial©…- ‡2 `*…bê„û€þ‚Œ Arial©…-…ð ‡2 `…u× ‡2 `(…uׄû ÿ‚Œ Arial©…-…ð ‡2 ÀÕ…1} ‡2 Àø…2}„û€þ‚Œ Arial©…-…ð ‡2 `…2ׄû€þ‚ŽPSymbol…-…ð ‡2 `î…+Ó„û€þ‚Œ Arial©…-…ð ‡2 `….k †& ÿ…û ‚¼"Systemn…-…ð…ð ˆÐÏࡱáN› hþO" See Also1 ÿ‹ Arial…-…ð †2  ÿ…û ‚¼Œ"Systemn…-…ð©2 ` ˆ1 ƒu ˆ2 †ÿ„Œ Arial…- „œw+…- „œw+ ‚ÿ‚À†Fÿ þÿÿ…+Ó ‡2 hc …+Ó‚ûƒEq…-…ðÿ&’ˆ3CompObj¨ ˆ2 ƒu ˆ3 †+Ë‚ „¸ ¯ lpT€$ =‚ …‚ÿ‚….ƒ1‚ ‡ € †&ÿƒ‰Àÿ¬ÿÀ , “& MathType0…ûéýゎPSymbol‚…-‡2 ùòƒ(…ûéýゎPSymbol‚…-…ð‡2 ù÷ƒ)…ûéýゎPSymbol‚…-…ð‡2 Âùƒ(…ûéýゎPSymbol‚…-…ð‡2 Âþƒ)„û€þ‚Œ Arial‚…-…ð ‡2 Û@…x¼ ‡2 Û‚…ti ‡2 Ûð…ti ‡2 Û¼…ti ‡2 ÛW …ti ‡2 ¤@…y¶ ‡2 ¤‰…ti ‡2 ¤÷…ti ‡2 ¤„…ti„û€þ‚ŽPSymbol…-…ð ‡2 ÛÕ…=Ó ‡2 Û¸…-Ó ‡2 ÛK …+Ó ‡2 ¤Ü…=Ó ‡2 ¤€…-Ó„û€þ‚Œ Arial‚…-…ð ‡2 Û…6Ö ‡2 Ûà…9Ö ‡2 Û{ …4Ö ‡2 ¤…4Ö ‡2 ¤¨…3Ö„ûàþ‚Œ Arial…-…ð ‡2 38…2  ‡2 3Î …3  ‡2 ün…3  ‡2 ü …2 „û€þ‚Œ Arial‚…-…ð ‡2 Ûu …,i ‡2 ¤§ ….i †& ÿ…û ‚¼"Systemn…-…ðƒ ¿ˆ1†+ƒtˆ1†+ƒt ˆ2R6DURP†îýPhšAónV†îýP‹Fø‹V…Àc†À‚ø…Àd½ebbie Johnson0\\WORDPUSH\MARKET¥3À^_ÉÊ jjjšfrÏ‹ø ÿtçÇFæ$ƒÿ(„…ð ‡2 `Õ¦=Ó ƒxƒt–(–)†=ˆ!£ü迎ÿÿ‹ø…ÿŒ¹ÿuøÿuü‹ËèÙ’þÿ‹øõP‚„°ÝO‚„ÄôÿÿƒÄ°„û€þ‚‰ Îôœôœ ‚ ÿƒ…x¼ ‡2 ²…tk ¥›ÇGº;ÇG"ë:öD@t3€dýÿtÿ‡2 f …y¼ ‡2 à ŸÒÒcMMÕôM¼ô¼ô¼M! ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½ÿ 3ÿ9=vçtTh8Ðæÿ̸ç9=vçt6‹lÄ » lp߃@< d‚ …‚ÿ‚….ƒ1‚ ‡ `†&ÿƒ‰Àÿ©ÿ © “& MathType…ûéýゎPSymbol‚…-‡2 þσ(…ûéýゎPSymbol‚…-…ð‡2 þÔƒ)…ûéýゎPSymbol…-…ð‡2 Þcƒ(…ûéýゎPSymbol…-…ð‡2 Þhƒ)…ûéýゎPSymbol…-…ð‡2 fƒ(…ûéýゎPSymbol…-…ð‡2 kƒ)„û€þ‚¼Œ Arial…-…ð ‡2 à@…f€„û€þ‚Œ Arial…-…ð ‡2 à_…ti ‡2 À±…x¼ ‡2 Àó…ti ‡2 ­…y¶ ‡2 ö…ti ‡2 àÞ …ti ‡2 àN…ti ‡2 à®…ti„û€þ‚ŽPSymbol…-…ð ‡2 ಅ=Ó ‡2 ºé…æ“ ‡2 3é…è“ ‡2 åé…ç“ ‡2 ºÞ…ö“ ‡2 3Þ…ø“ ‡2 åÞ…÷“ ‡2 àþ…=Ó ‡2 ß5 …æ“ ‡2 5 …è“ ‡2 5 …ç“ ‡2 ߺ …ö“ ‡2 º …ø“ ‡2 º …÷“ ‡2 à§ …+Ó ‡2 Ç…-Ó ‡2 …-Ó ‡2 ßÏ …æ“ ‡2 Ï …è“ ‡2 Ï …ç“ ‡2 ß*…ö“ ‡2 *…ø“ ‡2 *…÷“ ‡2 àÝ…+Ó ‡2 ß…æ“ ‡2 …è“ ‡2 …ç“ ‡2 ߊ…ö“ ‡2 Š…ø“ ‡2 Š…÷“„û€þ‚Œ Arial…-…ð ‡2 Çò …6Ö ‡2 ï …0Ö ‡2 Çb…9Ö ‡2 b…3Ö ‡2 ÇÂ…4Ö ‡2 Â…4Ö„ûàþ‚Œ Arial…-…ð ‡2 8Ê…2  ‡2 8%…3 „û€þ‚Œ Arial…-…ð ‡2 àÌ….i †& ÿ…û ‚¼"Systemn…-…ðƒb…9Ö ‡2 b„Ü…=Ó ‡2 ¤€…-Óƒ  ‡2 ün…3  ‡2 ü ÿ“ MathType0…ûéýãÿ&ƒ ˆÐÏࡱá ÿƒ…ð ‡2 Û@…x¼ „ûàþ‚Œ Arialÿn NativeˆÐÏࡱá.‘>CompObj ÿCompObjÿ!ÿ…Ö„ûàþ‚” Aria Arial‚…-…ð ƒ(…ûéýょPSy „…ßÌ ‚@‚Œ Arial‚…-†bol‚…-…ð…2 Âùƒ(…ûéýゎPSybol‚…-…ð†2 Âþƒ(…ûéýゎPSyal‚…-…ð 2 Û@al‚…-…ð …2 Û@‹Native‡2 ÛW …ti ‡2 ¤@y ‡2 ¤‰…ti ‡2 ¤÷ ‡2 ¤„…ti„û€þ…-…ð‡2 Âùƒ(‡2 Û{ …4Ö ‡2 ¤…4ÿ…û ‚¼…"ÿƒ„û€þ‚ Arialÿƒ‰Àÿ¬ÿÀ , ƒ&ƒÖ ‡2 Ûà…9Ö †2 Û{ ‚ ‡ € ‰&¨…3Ö„ûàþ‚‡META† 38…2  ‡2 3Î …3 ÿ‰2 ¤Ü…=Ó ‡2 ¤€-‚‚…-…ð ‡2 Ûu  ‡PIC “ÿuÿ(K‰E؃}Ø„<ˆhJ¸jj…-…ð ‡2 Ûu ,…ti ‡2 Û¼…ti ‚2 "Œ2 ìðVWè£ÿ…À„sèÆYþÿ‹ø¥8 Rÿ 8 PSymbol‚…-‡2 ùòG‚Œ Arial‚-ÿ M=0x4 WINLINK=0x20­ ˆ3 ‚,ƒyƒt–(–hType0…ûéýゃ!Ÿ~Wšè&¯1€~ðtFðPÿv šÓH5 Àtƒœ-cƒ(…û•ýâ‚…Ÿebbie Johnson0\\WORDPUSH\MA0 DS Equation ‰Equation.(    “ŠlpSž€Ú   ‚ …‚ÿ‚….ƒ1‚ ‡ `@†&ÿƒ‰Àÿ®ÿ “& MathType`…ûþ傎PSymbol‚…-‡2 ³½ƒ(…ûþ傎PSymbol‚…-…ð‡2 ³½ƒ)„û€þ‚¼Œ Arial©…-…ð ‡2  ;…f~ ‡2  ã…a× ‡2  í…a× ‡2  " …a× ‡2  …aׄû€þ‚Œ Arial‚…-…ð ‡2  I…tk ‡2  <…tk ‡2  ¥ …tk ‡2  ‹…tk„û ÿ‚Œ Arial©…-…ð ‡2 ù…n} ‡2 ô…n}„û€þ‚ŽPSymbol…-…ð ‡2  ¡…=Ó ‡2  Ç…+Ó ‡2  ü…+Ó ‡2  $ …+Ó ‡2  ï…+Ó„û ÿ‚Œ Arial©…-…ð ‡2 Í…0} ‡2 Ê…1} ‡2  …2} ‡2 ô) …2}„û€þ‚Œ Arial…-…ð ‡2  O….k ‡2  ½….k ‡2  +….k ‡2  ›….k †& ÿ…û ‚¼"Systemn…-…ð„û€þ„º …÷“ ‡2 à§ …+Ó ‚…-…ð ‡2 àÌ ‡2 ßÏ …æ“ ‡2 Ï …è“ ‡2 Ï …ç“ „2 ß!† *…÷“ ‡2 à݃+Ó‡2 ß…æ“ ‡2 $„Â…4Ö„ûàþ‚ƒÓ ‡2 ß5 …æ“ “2 5  Arial…-…ð † Çò …6Ö ‡2 ï ƒ0Ö‡2 Çb…9Ö ‡2 bƒÖ ‡2 ÇÂ…4Ö †2 Â…è“ ‡2 5 …ç“ ‹2 rial…-…ð ‡2 8…2  ‡2 8%…3 ƒ€þ‚Œ Arial† ºÞ…ö“ ‡2 3Þƒø“†& ÿ…û ‚ŽPSymbol‚ ƒb…9Ö ‡2 b%ƒi„û€þ‚‰PSymbaXlp <@v  þ‚ …‚ÿ‚….ƒ1‚ ‡ À †&ÿƒ‰Àÿ¿ÿÀ  Ž& MathType‚…ûþ傎PSymbol‚…-‡2 Šùƒ(…ûþ傎PSymbol‚…-…ð‡2 Šùƒ)…ûþ傎PSymbol‚…-…ð‡2 úƒ(…ûþ傎PSymbol‚…-…ð‡2 úƒ)„û€þ‚Œ Arial©…-…ð ‡2 w:…x¼ ‡2 w……tk ‡2 w™…tk ‡2 9…y¼ ‡2 †…tk ‡2 H…tk ‡2 n…tk„û€þ‚ŽPSymbol…-…ð ‡2 w×…=Ó ‡2 w9…-Ó ‡2 Ø…=Ó ‡2 …-Ó„û€þ‚Œ Arial©…-…ð ‡2 w…2× ‡2 wb…2× ‡2 …2× ‡2 7…2ׄû ÿ‚Œ Arial…-…ð ‡2 ^æ…2}„û€þ‚Œ Arial©…-…ð ‡2 w÷…,k ‡2 s ….k †& ÿ…û ‚¼"Systemn…-…ð‡©n…-…ð„û€þ‚…n} ‡2 ô…n}‚…-…ð ‡2 àÌ ‡2 ßÏ …æ“ ‡2 Ï ƒ ŒÀŽ˜‚jðð÷æ!† *…÷“ ‡2 à݃+Ó…‚ÿ‚….‰1  I…tk ‡2  <ƒtk †èèƒÓ ‡2 ß5 …æ“ “2 5  Arial…-…ð † Çò …6Ö ‡2 ï ƒ0Ö‡2 Çb…9Ö ‡2 b'…è“ ‡2 5 …ç“ ‹2 rial…-…ð ‡2 8…2  ‡2 8%…3 L† ºÞ…ö“ ‡2 3Þ‰ø“…aׄû€þ‚‚ŽPSymbol‚ ƒb…9Ö ‡2 b ¤Àÿt+ƒ=¤Àt"ÿ ¤ÀuSjSÍ }ÿ6 - lp¢ƒÀ  ß‚ …‚ÿ‚….ƒ1‚ ‡  †&ÿƒ‰Àÿ©ÿ`© “& MathType…ûéýゎPSymbol…-‡2 Þ¯ƒ(…ûéýゎPSymbol…-…ð‡2 Þ»ƒ)…ûéýゎPSymbol…-…ð‡2 ´ƒ(…ûéýゎPSymbol…-…ð‡2 Àƒ)„û€þ‚Œ Arial…-…ð ‡2 Àù…x¼ ‡2 À?…ti ‡2 õ…y¶ ‡2 D…ti ‡2 à“ …ti ‡2 àË…ti„û€þ‚ŽPSymbol …-…ð ‡2 º1…æ“ ‡2 31…è“ ‡2 å1…ç“ ‡2 º3…ö“ ‡2 33…ø“ ‡2 å3…÷“ ‡2 àY…=Ó ‡2 ß–…æ“ ‡2 –…è“ ‡2 –…ç“ ‡2 ß…ö“ ‡2 …ø“ ‡2 …÷“ ‡2 à$…+Ó ‡2 Ç …-Ó ‡2 ßE …æ“ ‡2 E …è“ ‡2 E …ç“ ‡2 ß¡ …ö“ ‡2 ¡ …ø“ ‡2 ¡ …÷“ ‡2 à\ …+Ó ‡2 =…-Ó ‡2 ß}…æ“ ‡2 }…è“ ‡2 }…ç“ ‡2 ßÙ…ö“ ‡2 Ù…ø“ ‡2 Ù…÷“„û€þ‚Œ Arial…-…ð ‡2 ÇP…2Ö ‡2 P…0Ö ‡2 ÇØ …2Ö ‡2 k …2Ö ‡2 Ç£…0Ö ‡2 …2Ö„ûàþ‚Œ Arial …-…ð ‡2 8X…2 „û€þ‚Œ Arial…-…ð ‡2 à ….i †& ÿ…û ‚¼"Systemn…-…ð ‚ ÿ ‚ÿþÿ    þÿþÿþÿþÿÿÿ!‚…=Ó ‡2 e,…-Óÿÿ(œL Œp É|…OleÿÿFp É|H†èè‚ ÿ2“ <&] þ‚ ‡2 …2× ‡2 7ƒ1‚ ‡ À …ÿƒ‰Àÿ¿ÿÀ  ƒ&(‚…-…ð 2 à2 n…tk„û€þ‡¼ ‡2 w……tk †2 w™…-…ð ‡2 ^æ2…ûþå‚‹PSymbolƒ(…ûþ傎PSybol‚…-…ð†2 úƒ)„û€þ‚… Ar‘ˆð÷ðoo_oo„ ‚.Ž& MathTyˆ™…tk ‡2 9…y¼ ‡2 †…tk ‡2 Ht ‡2 n…tk„û€þ“0-00000 0Equa ‡2 w×…=Ó ‡2 w9ÿƒk †èèŒ Arial©…-…ð († ºÞ…ö“ ‡2 3ÞÊ Á lp­ ‹€H ‰‚ …‚ÿ‚….ƒ1‚ ‡ À€ †&ÿƒ‰Àÿ²ÿ@ r “& MathType°…ûéýゎPSymbol‚…-‡2 µöƒ(…ûéýゎPSymbol‚…-…ð‡2 µ4ƒ)…ûéýゎPSymbol‚…-…ð‡2 ~ÿƒ(…ûéýゎPSymbol‚…-…ð‡2 ~=ƒ)…ûéýゎPSymbol‚…-…ð‡2 õƒ(…ûéýゎPSymbol‚…-…ð‡2 3ƒ)„û€þ‚Œ Arial‚…-…ð ‡2 —@…x¼ ‡2 —|…sÀ ‡2 —¸…ti ‡2 —×…ti ‡2 `@…y¶ ‡2 `……sÀ ‡2 `Á…ti ‡2 `…ti ‡2 ` …ti ‡2 ë6…zÀ ‡2 ë{…sÀ ‡2 ë·…ti ‡2 ëM…sÀ ‡2 — …,i ‡2 —8 …,i ‡2 `)…,i ‡2 `è …,i ‡2 ë…,i ‡2 ëñ….i„û€þ‚ŽPSymbol…-…ð ‡2 —…=Ó ‡2 —s…-Ó ‡2 `…=Ó ‡2 `^…-Ó ‡2 ë…=Ó„û€þ‚Œ Arial‚…-…ð ‡2 —N…2Ö ‡2 —Ÿ…2Ö ‡2 `W…2Ö ‡2 `Š…2Ö„ûàþ‚Œ Arial…-…ð ‡2 ¸C …2  †& ÿ…û ‚¼"Systemn…-…ð‡2 —Ÿ2#ƒÿ‚Àÿ„ ÿ!‚ýゎPSymbol‚‚-‚ÿþÿ    Œ ˆ2 ‚.z|c„}‡tiveÿ!ÿ„ ÿ®)†=ˆ2˜ƒt†-ˆ2˜ƒt 0B+Ò&â"…ObjÿÿŸƒxƒt–(–)ƒy‚Œ Arial‡-ÿÿÿÿ ‡2 ß}…æ“ ‡2 }†èèJ…–(–ÿ"‚¼"Systemnƒ-‰)» ù…x¼ ‡2 À?…ti MathType…ûéýã‚CompObjƒ(…ûéýã‚PSy=ˆ2ˆ0„CŸö±cÄ­ý·øÑ¹òø2iL–>jÿÛøø'óâ™zƒ(…ûéýゎPSybol…-…ðŒ2 Àÿƒ‰Àÿ©ÿ`© ‹&al…-…ð …2 Àù…x¼ ‡2 À?…ti ‚2 b ‡2 ß…ö“ 2 P…0Ö ‡2 ÇØ …2Ö ƒ“ ‡2 º3…ö“ †2 33ÿ‡2 àY…=Ó ‡2 ß–æ ‡2 –…è“ ‡2 –B† à$…+Ó ‡2 Ç ƒ-Óƒƒ“ ‡2 E …ç“ 2 ß¡ -…ð ‡2 º1‰æ“¡ …÷“ ‡2 à\ “+Ó "Systemn…-…ðŠhType…ûéýりö“ ‡2 33…ø“ „2 å L ˆp Ƀ¼ ‡2 À?…ti †2 õ1 ( lp4)y À ù‚ …‚ÿ‚….ƒ1‚ ‡ € `%†&ÿƒ‰Àÿºÿ %: “& MathType0…ûþ傎PSymbol‚…-‡2 c(…ûþ傎PSymbol‚…-…ð‡2 ïñƒ)…ûþ傎PSymbol…-…ð‡2 ¯…ƒ(…ûþ傎PSymbol…-…ð‡2 ¯¹ƒ)…ûþ傎PSymbol…-…ð‡2 ï…ƒ(…ûþ傎PSymbol…-…ð‡2 ﹃)…ûþ傎PSymbol…-…ð‡2 /ƒ(…ûþ傎PSymbol…-…ð‡2 /³ƒ)„û€þ‚¼Œ Arial…-…ð ‡2 Ü;…f~ ‡2 Ï …a× ‡2 ψ…a× ‡2 Ïæ …a× ‡2 Ïæ…a× ‡2 ÏÇ…a× ‡2 ÏÛ…a× ‡2 ϧ…aׄû€þ‚Œ Arial…-…ð ‡2 ÜA…sÀ ‡2 Ü}…tk ‡2 œÆ…x¼ ‡2 œ …sÀ ‡2 œE…tk ‡2 ÜÄ…y¼ ‡2 Ü …sÀ ‡2 ÜE…tk ‡2 Æ…z¼ ‡2 …sÀ ‡2 ?…tk ‡2 Ï£ …sÀ ‡2 Ïò…tk ‡2 ÏÝ…sÀ ‡2 ÏÓ…tk ‡2 Ïñ…sÀ ‡2 Ï …tk ‡2 Ï­!…sÀ ‡2 ÏJ#…tk ‡2 Üî…,k ‡2 œ¶…,k ‡2 ܶ…,k ‡2 °…,k ‡2 ÏÕ#….k ‡2 ÏH$….k ‡2 Ï»$….k„û€þ‚ŽPSymbol …-…ð ‡2 ÜÏ…=Ó ‡2 ©…æ‘ ‡2 <…è‘ ‡2 Ô…ç‘ ‡2 G…ç‘ ‡2 a…ç‘ ‡2 ©- …ö‘ ‡2 <- …ø‘ ‡2 Ô- …÷‘ ‡2 G- …÷‘ ‡2 a- …÷‘ ‡2 ÏÏ…=Ó ‡2 Ï[…+Ó ‡2 Ϲ …+Ó ‡2 Ϲ…+Ó ‡2 Ïš…+Ó ‡2 Ï®…+Ó ‡2 Ïz…+Ó„û ÿ‚Œ Arial…-…ð ‡2 / é‡00}} ‡2 / Y ‡10}} ‡2 / Ä ‡01}} ‡2 / ·‡11}} ‡2 / §‡20}} ‡2 / ¹‡02}} ‡2 / ‡ ‡21}} ‡2 #¸…2} ‡2 #„…2} ‡2 #t"…2} †& ÿ…û ‚¼"Systemn…-…ð…÷‘ ‹2 Ïð ‡2 —@…x¼ „2 —*­ ‹0؆èèP ‡2 `Á…ti ‡2 `„ûàþ‚Œ Arialƒ1‚ ‡ À€ …ÿƒ‰Àÿ²ÿ@ r ƒ&Œxï{xxxxŽ(–)†=ƒs‚.…ol‚…-‡2 ~=ƒ)…ûéýã‚:‚(†bol‚…-…ð…2 ~ÿ‡2 `…=Ó ‡2 `^†bol‚…-…ð†2 ~="…-…ð ‡2 —Nƒ2Ö ÿ†bol‚…-…ð†2 3ƒ)„û€þ‚Œ Ar —…=Ó ‡2 —sƒ-Ó…x¼ ‡2 —|…sÀ ‡2 ¸…ti ‡2 —×…ti "ƒn Equation.2ô9²q&‡‚…-…ð‡2 õ‡2 `)…,i ‡2 `è ‚‚…-…ð‡2 ~ÿŒ Arial‚…-‚ð…,i ‡2 ë…,i ‡2 ñ….i„û€þ‚Šystemn…-…ð …ûéýã‚‹PSymbol‡2 `…=Ó ‡2 `^‡2 ë…=Ó„û€þ ‡2 ` …ti ‹2 ë6‚…-…ð‡2 µ4‚…2Ö ‡2 `W…2Ö $¨ƒt–(–)†=ˆ2˜ƒt†-ˆ2˜2 —8 …,i ‡2 `),    ÿƒ‰Àÿ²ÿ@ r ˆ& Maú ñ lpí(<€¨ ‚ …‚ÿ‚….ƒ1‚ ‡ À %†&ÿƒ‰Àÿ¶ÿà$v Ž& MathType‚…ûþ傎PSymbol~…-‡2 Ó4ƒ(…ûþ傎PSymbol~…-…ð‡2 Ó4ƒ)…ûþ傎PSymbol~…-…ð‡2 ³Jƒ(…ûþ傎PSymbol~…-…ð‡2 ³Jƒ)…ûþ傎PSymbol~…-…ð‡2 óKƒ(…ûþ傎PSymbol~…-…ð‡2 óKƒ)„û€þ‚ŽPSymbol<…-…ð ‡2 ÄÓ…¢` ‡2 À…=Ó ‡2 ¤é…¢` ‡2 äê…¢` ‡2 ­U…æ‘ ‡2 U…è‘ ‡2 ØU…ç‘ ‡2 ­¿…ö‘ ‡2 ¿…ø‘ ‡2 Ø¿…÷‘ ‡2 Àå…=Ó ‡2 º …-Ó ‡2 ºÀ…+Ó ‡2 ú·…-Ó ‡2 ¯" …æ‘ ‡2 þ" …è‘ ‡2 Ú" …ç‘ ‡2 ¯ä…ö‘ ‡2 þä…ø‘ ‡2 Úä…÷‘ ‡2 À …=Ó ‡2 ÊG…æ‘ ‡2 ãG…è‘ ‡2 õG…ç‘ ‡2 ÊÊ…ö‘ ‡2 ãÊ…ø‘ ‡2 õÊ…÷‘ ‡2 ÀÜ…+Ó ‡2 ŸÅ…-Ó ‡2 ß1…-Ó ‡2 Ê…æ‘ ‡2 ã…è‘ ‡2 õ…ç‘ ‡2 Ê=…ö‘ ‡2 ã=…ø‘ ‡2 õ=…÷‘ ‡2 Àÿ…+Ó ‡2 Ê' …æ‘ ‡2 ã' …è‘ ‡2 õ' …ç‘ ‡2 Êj"…ö‘ ‡2 ãj"…ø‘ ‡2 õj"…÷‘„û€þ‚¼Œ Arial~…-…ð ‡2 À;…f~„û€þ‚Œ Arial©…-…ð ‡2 ÀÀ…tk ‡2  …x¼ ‡2  Ö…tk ‡2 à…y¼ ‡2 à×…tk ‡2 ºù…tk ‡2 ºº…tk ‡2 ú2…tk ‡2 úð…tk ‡2 À8…tk ‡2 Àe#…tk ‡2 º× …6× ‡2 º ‡18×× ‡2 ºÐ‡12×× ‡2 úH ‡12×× ‡2 úÝ…6× ‡2 Ÿü…6× ‡2 ßý…0× ‡2 Ÿ˜‡18×× ‡2 ß…6× ‡2 ŸÅ ‡12×× ‡2 ßÅ ‡12×ׄû ÿ‚Œ Arial~…-…ð ‡2 >…2} ‡2 N¶…2} ‡2 é#…2}„û€þ‚Œ Arial©…-…ð ‡2 À‚$….k †& ÿ…û ‚¼"Systemn…-…ð‡12×× ‚2 µþèˆh ‡2 ˜1…ç“ ‡2 ºc…‚ÿ‚….ƒ1‚ ‡ @@ÿƒÖ ‡2 À…2Ö ‰2 ˜ …ûéýã‚‹PSymbol„  ÿƒ)…ûéýょPSy…ø“ ‡2 åc…÷“ ‚2 ƒ(…ûéýょPSyfƒO"†bol…-…ð†2 ðƒ(…ûéýゎPSybol…-…ð2 ^ì2 ƒ…=Ó ‡2 ¼ ¢-”;V–>Y™A[@\=X›lp&WÀB 7‚ …‚ÿ‚….ƒ1‚ ‡  †&ÿƒ‰Àÿ¼ÿÀÜ “& MathType`„û€þ‚Œ Arial©…-‡2 `. ŸSo the der×kk××k××~‡2 `! Ÿivatives aT¼×kT¼×Àk× ‡2 `\ t (skk~À ‡2 `% arek×~ׄû ÿ‚Œ Arial©…-…ð ‡2 Àu…0}„û€þ‚Œ Arial©…-…ð ‡2 `…,k ‡2 `¥…)~ ‡2 `…tk„û ÿ‚Œ Arial©…-…ð ‡2 À…0} †& ÿ…û ‚¼"Systemn…-…ð‚ˆ Arial…æ‘ ‡2 U…è‘ ‡2 U…ç‘ ‡2 ­¿…ö‘ ‡18×× ‡2 ß…6× …2 ‡2 Àå…=Ó ‡2 º …-Ó ‡2 ºÀ…+Ó „2 ú‹  þ" …è‘ ‡2 Ú" „ç‘ÖÍlp0Œ@`  7‚ …‚ÿ‚….ƒ1‚ ‡  €†&ÿƒ‰Àÿ¶ÿ@Ö “& MathTypeà …úƒ"…-‡ý@‡ý…ûæýÛ‚ŽPSymbol‚…-‡2 Mƒ(…ûæýÛ‚ŽPSymbol‚…-…ð‡2 Ƀ)‡ý÷ ‡ý …ûæýÛ‚ŽPSymbol‚…-…ð‡2 Ô ƒ(…ûæýÛ‚ŽPSymbol‚…-…ð‡2 Pƒ)„û€þ‚ŽPSymbol…-…ð ‡2 ho…¶¼ ‡2 ŒW…¶¼ ‡2 h …¶¼ ‡2 Œ …¶¼„û€þ‚¼Œ Arial‚…-…ð ‡2 hf…f~ ‡2 h …f~„û€þ‚Œ Arial©…-…ð ‡2 ŒF…sÀ ‡2 `Ñ…sÀ ‡2 `±…tk ‡2 Œ …tk ‡2 `X …sÀ ‡2 `8…tk„û ÿ‚Œ Arial‚…-…ð ‡2 À–…0} ‡2 À'…0} ‡2 À …0} ‡2 À®…0}„û€þ‚Œ Arial©…-…ð ‡2 `"…,k ‡2 `© …,k ‡2 `ä….k‡2 `Q‘ and k×××k †& ÿ…û ‚¼"Systemn…-…ð‚’ Arial©OleUƒ ‡2 Êj"…ö‘ ‡2 ãj"ø#ƒÿ œ-‡2 `R “tives at (kT¼×À‚ ÿ$‚ˆÿþÿn Native „<á‚×kk~ ‡2 `ú …sÀ ÿÿÿ‹ Arial©…-‡2 `. ÿ‹ek×~ׄû ÿ‚‚ÿÿƒ ‡2 `¥…)~ ‡2 `“0-00000 0ˆEqua lpf¯ÀÊ  R‚ …‚ÿ‚….ƒ1‚ ‡ @à†&ÿƒ‰Àÿ¯ÿ ï “& MathTypeð …úƒ"…-‡Rô ‡ô ‡R¾ ‡¾ ‡Rï ‡ï ‡R¹ ‡¹ ‡ý– ‡ý„û€þ‚¼‰ Arial…- ‡2 `&…N ‡2 hK …N ‡2 ŒK …N„û€þ‚‰ Arial…-…ð‡2 `: Ÿ (normalizk~××~B×TT¼ ‡2 `Ì‹ed)××~„û€þ‚ŽPSymbol…-…ð ‡2 `P …=Ó„û€þ‚‰ Arial…-…ð ‡2 `A….k †& ÿ…û ‚¼"Systemn…-…ðšã–­¥¥Ï{ŠRE)Ç9 c0„0„ÿÿšÖ‚Œ Arial~ƒ-…ð ‡2 `~ …,i ‚2 ‡2 `~ …,i ‡2 `V ƒ€ ‡2 `­…,i †2 `€…,i ‡2 `µ!….i ‡2  …ti ‡2 Œ¿‰sÀ …ti ‡2 ŒÈ…ti „Û …0 „û€þ‚…sÀ ‡2 ` …ti„ûàþ‡2 `­…,i ‡2 `€„Œ…ûéýã‚PSymbti ‡2 ŒÈ…ti „2 `"„û€þ‚šPSymbolCompObjƒ¼ ‡2 ŒÍ…¶¼ †2 j¾…¶¼ ‡2 ŒÌ…¶¼„ûþ‚¼Œ Arial~‚-!† jº…f€ †& ÿ%‡2 `8…tk„û ÿ‚H8ƒ ÿƒÀ‚F„MicrŽPSymbol‚…-ƒð"“soft Equation 2.0 ŠDS Equatiÿÿÿ!ƒn Equation.2ô9²q ÿÿÿ…þÿÿ‡2 ŒF…sÀ ‡2 `ÑŸ6FA7DF;KJ@MG é‚ …‚ÿ‚….ƒ1‚ ‡ € †&ÿƒ‰Àÿ©ÿÀ ) “& MathTypep…ûéýゎPSymbol‚…-‡2 ¾ƒ(…ûéýゎPSymbol‚…-…ð‡2 ¾Œƒ)…ûéýゎPSymbol‚…-…ð‡2 ¾œƒ(…ûéýゎPSymbol‚…-…ð‡2 ¾¤ƒ)„û€þ‚¼Œ Arial§…-…ð ‡2  $…li ‡2  =…bé ‡2   …bé„û€þ‚Œ Arial§…-…ð ‡2  …ti ‡2  )…ti ‡2  s …ti„û€þ‚ŽPSymbol…-…ð ‡2  j…=Ó ‡2  ó…-Ó ‡2  < …+Ó„û€þ‚Œ Arial§…-…ð ‡2   …1Ö„ûàþ‚Œ Arial§…-…ð ‡2 *…0  ‡2 è …1 „û€þ‚Œ Arial§…-…ð ‡2  d …,i †& ÿ…û ‚¼"Systemn…-…ðð öt ÿ6zIš Wÿÿÿ„ÿ‚ÀF ‰ÀXœÐ~º‚À MathType‚ …úˆ ƒ‚ h@" Normal Indent †&ÿƒ‰ÀÿÀÿÀ  „ÿÿ ƒÿ嗈鞋x‡‡METAþÿÿþÿ(œãaÿÿã–­¥Ï{ŠRE)Ç9 c0„ÿÿ‚ˆ' Equation 2.0 ŽDS EqComÿþÿþÿÿ$‡2 `A….k †& ÿƒ~‰°àþÿ< ´ hþO" TablePaalized)†=!ÿ„5ÿÿ!ÿ]À„  À•À@€   R¾ ‡¾ ‡Rï „™D‚xtD‚DOD d mode.StopCancel docum…‚ÿ‚ÿ..(%Open a document on the Wortting HtmlResIWr,*Create dcument with InternetWorks objec. Bookmark@>Create §a bookmrk name for a Hyperlink to a bo†°àþÿ ¾HtmlDirectInput HTML Marku.List all URLs in document.!‚ÿ‹HcCÿŽXgGÿþ‚ŽPSymbol‹-vVÿ‹`xXÿ‡PyÿÿŸoBackGo back one jump. „û€þ‚‹ Arial!Ÿt RESET button in form. Inse‚¼"Systemn¥-w all places visited recently.‡ý„û€þ‚¼ƒ úlp €º  ‚ …‚ÿ‚….ƒ1‚ ‡  `†&ÿƒ‰Àÿ¡ÿ A “& MathTypeð„û€þ‚¼Œ Arial©…- ‡2 À*…bê ‡2 ÀÙ …bê„û ÿ‚Œ Arialn…-…ð ‡2 '…0} ‡2 É …1}„û€þ‚Œ Arial©…-…ð‡2 À~‘ and k×××k„û€þ‚ŽPSymbol…-…ð ‡2 À<…=Ó ‡2 Ây…æ‘ ‡2 ëy…è‘ ‡2 íy…ç‘ ‡2 ´…ö‘ ‡2 ë´…ø‘ ‡2 í´…÷‘ ‡2 Àº …=Ó ‡2 Ä÷ …æ‘ ‡2 é÷ …è‘ ‡2 ï÷ …ç‘ ‡2 Ä…ö‘ ‡2 é…ø‘ ‡2 ï…÷‘„û€þ‚Œ Arial©…-…ð ‡2 –9…x¼ ‡2 Ö7…y¼ ‡2 ™¶ …x¼ ‡2 Ùµ …y¼„û ÿ‚Œ Arial…-…ð ‡2 ö…0} ‡2 6…0} ‡2 ù…1} ‡2 9€…1}„û€þ‚Œ Arial©…-…ð ‡2 ÀÒ…,k †& ÿ…û ‚¼"Systemn…-…ð…-ÿ„ÿ‚ÀF! MathType‚ …úˆ ƒ‚ h@" Normal Indent †&ÿƒ‰ÀÿÀÿÀ  „ÿÿ ƒÿƒÿ ÿÿPSymbol…-ð ystemn…-…ð‰ð ð ‡2 *…0  ’2 ÿ÷½à{à{à{à{à{‘÷½÷½ôœôœôœôœ ‡2  < …+Ó„û€þÿþÿþÿ†&ÿƒˆÀÿÀÿÀ !‚…-…ð ‡2  $ƒ~‰°àþÿ< ™ hþO" TablePad …,i †& ÿ%:1lp¬¯€( ü‚ …‚ÿ‚….ƒ1‚ ‡ @ †&ÿƒ‰Àÿ»ÿàû “& MathTypeà…ûéýゎPSymbol…-‡2 žƒ(…ûéýゎPSymbol…-…ð‡2 žŒƒ) …úƒ"…-‡°‡:‡u ‡% „û€þ‚¼Œ Arial§…-…ð ‡2 €$…li ‡2 ‹m…bé ‡2 € …bé„û€þ‚Œ Arial§…-…ð ‡2 €…ti ‡2 ‹Ð…ti ‡2  …ti„û€þ‚ŽPSymbol …-…ð ‡2 €j…=Ó ‡2 €›…+Ó ‡2 €D …-Ó ‡2 ¨¾…æ“ ‡2 …¾…è“ ‡2 ª¾…ç“ ‡2 ¨. …ö“ ‡2 …. …ø“ ‡2 ª. …÷“„ûàþ‚Œ Arial§…-…ð ‡2 ëd…0  ‡2 àõ …1 „û€þ‚Œ Arial§…-…ð ‡2 §‡10ÖÖ ‡2 €_…1Ö ‡2 §r ‡10ÖÖ ‡2 €{…,i †& ÿ…û ‚¼"Systemn…-…ðƒ…ð ‡2 à…1  ©2 B祋ÌÒ^_ÉÊU‹ìƒ>~FPjèÏ …,i †& ÿ„ ‚¼"Systemnÿ„ÿÿL†þÿ      ‡METAÿ„ ÿ ÿÿ fÿ‡Obj‰Entryÿÿ…è“ ‡2 æÿ…ç“ ‚2 ‡2   …1Ö„ûàþ‹ é‚ ÿ>ƒ“ ‡2 »r…ö“ 2 òrbol‚…-…ð‡2 ¾œÿ     ƒ%…U‚ …‚ÿ‚EdFi]™  AEquationÿƒ1‚ ‡  `ü MathTypeð„û€þ‚¼ƒ ÿ‚Œ Arialÿÿ‚Œ Arialn…-…ð ‡2 '…0} ›EdFi]—  AEquationˆ ƒ‚ …x¼ ‡2 Ö7…y¼ …2 ™ ÿ„ÿ‚‚Œ Arial©…-‡2 Ây…æ‘ ‡2 ëy†è ÿ‚Œ Arial…ö‘ ‡2 ë´…ø‘ „2 íÿÿ‡ Ä÷ …æ‘ ‡2 é÷ ‹è‘œôœ ‡2  < …+Ó‰û€þ‘ ‡2 é…ø‘ †2 ï…÷‘„û€þ‚Ž "Systemn…-…ðÿƒ‰Àÿ¡ÿ A ˆ& MaßÖlp¬ÁÀr Ï‚ …‚ÿ‚….ƒ1‚ ‡ € †&ÿƒ‰Àÿ©ÿà) “& MathTypep…ûéýゎPSymbol~…-‡2 ¾ƒ(…ûéýゎPSymbol~…-…ð‡2 ¾Òƒ)…ûéýゎPSymbol~…-…ð‡2 ¾êƒ(…ûéýゎPSymbol~…-…ð‡2 ¾ü ƒ)„û€þ‚¼Œ Arial§…-…ð ‡2  $…li ‡2  Ú…bé ‡2  …li ‡2   …bé„û€þ‚Œ Arial¨…-…ð ‡2  …0Ö ‡2  W ‡10ÖÖ„ûàþ‚Œ Arial§…-…ð ‡2 ·…1  ‡2 ñ …0 „û€þ‚ŽPSymbol…-…ð ‡2  °…=Ó ‡2  Ú …=Ó„û€þ‚Œ Arial§…-…ð‡2  9‘ and iÖÖÖi ‡2  —….i †& ÿ…û ‚¼"Systemn…-…ð† €{…,i †& ÿ ÿ„ ‡2 à…1  ¨2 B祋ÌÒ^_ÉÊU‹ìƒ>~FPjèψþÿ ÿƒÀF…,i †& ÿ„ ‚¼"Systemnÿ„ÿEquation.2ô9²q ‚ÐÏL†þÿ      ‡METAÿ‡-…ð ‡2 ëdƒ0 ÿ fÿ‡Obj‰Entryƒi„û€þ‚ŠPSymbÿÿ ƒt˜b ˆ0 ˆ1ˆEquatiÿa¦î˜À꘽ã¡Âã¦Ãâ¦Âà¤ÀÞ®ÌéžÁâ½àœ¼ßbol‚…-…ð‡2 ¾œÿJƒ%…U‚ …‚ÿ‚£EdFi]™  AEquationbol…-…ð¦2 žŒ­¶µ÷½YÎyÎ8Æ÷½÷½XÆ8Æ–µ•­÷½8ƾU­$‡2 €j…=Ó ‡2 €›$ÿ‡°‡:‡u ‘MathTypeð„û€þ‚¼ƒ ÿ‚Œ Arialƒ§…-…ð ‡2 €$‚ ‡ @ ƒ&"Ÿ­?X¨6M2E–2@’.9,5~M ; 8@"…x¼ ‡2 Ö7…y¼ „2 ™"‚Œ Arial©…-†°‡:‡u † ÿ‚Œ Arialÿ„ ÿÿÿ‡ Ä÷ …æ‘ ‡2 é÷ ƒè‘#‰mpObj3*lp¾À ß‚ …‚ÿ‚….ƒ1‚ ‡ À` †&ÿƒ‰Àÿ®ÿ n “& MathType€…ûýゎPSymbol…-‡2 ÷ºƒ{…ûýゎPSymbol…-…ð‡2 ÷ ƒ}„û€þ‚Œ Arial…-…ð ‡2 À…P ‡2 Àc…tk ‡2 ÀY…tk ‡2 À…tk ‡2 À …tk„û ÿ‚Œ Arial…-…ð ‡2 … …kp„û€þ‚ŽPSymbol…-…ð ‡2 À„…=Ó„û€þ‚Œ Arial…-…ð ‡2 ÀI…1ׄû ÿ‚Œ Arial…-…ð ‡2 Ý…2} ‡2 ’…3}„û€þ‚Œ Arial…-…ð ‡2 ÀË…,k ‡2 ÀÁ…,k ‡2 Àv…,k ‡2 À*…,k ‡2 À¦….k ‡2 À" ….k ‡2 Àž ….k ‡2 Àn …,k ‡2 À¼ …,k †& ÿ…û ‚¼"Systemn…-…ð  ÿ„ ‡2 à…1  È2 B祋ÌÒ^_ÉÊU‹ìƒ>~FPjèÏ ˆ1  and lˆþÿ ÿ… ÿ$ˆ¬ÿEquation.2ô9²q ‚Ðχ2  W ‡10ÖÖ„ûàþ0†þÿ      CompObjÿ‡-…ð ‡2 ëdƒ0 ÿ fÿ‡Obj „ ‚¼‘"Systemni„û€þ‚ŠPSymbÿ‚ÿ…û ‚¼žƒt˜b ˆ0 ˆ1ˆ‹Equatiƒ „û€þ‚ŒPSymb„û€þ‚¼ˆ Arial…ûéýã‚‹PSymbol‚ †þ‚Œ Arial¨¥-ˆ1ˆ0–(–)†=b „û€þ‚¼ˆ ArialJ“soft Equation 2.0 ŒDS Equati‚ …‚ÿ‚"†bol…-…ð©2 žŒ­¶µ÷½YÎyÎ8Æ÷½÷½XÆ8Æ–µ•­÷½8ƾU­~…-…ð‡2 ¾Ò‡2 €j…=Ó ‡2 €›D;lpÄä< k‚ …‚ÿ‚….ƒ1‚ ‡  `†&ÿƒ‰Àÿ¹ÿ Y “& MathTypep…ûþ傎PSymbol~…-‡2 Ó½ƒ(…ûþ傎PSymbol~…-…ð‡2 Ó½ƒ)…ûþ傎PSymbol~…-…ð‡2 Ó©ƒ(…ûþ傎PSymbol~…-…ð‡2 Ó¬ƒ)…ûþ傎PSymbol~…-…ð‡2 Óüƒ(…ûþ傎PSymbol~…-…ð‡2 Óÿƒ)„û€þ‚¼Œ Arial©…-…ð ‡2 À;…f~ ‡2 Àã…bê ‡2 Ày …bê ‡2 Àú…bê„û€þ‚Œ Arial~…-…ð ‡2 ÀI…tk ‡2 À8…tk ‡2 ÀW…tk ‡2 À‹…tk ‡2 À´…tk„û€þ‚ŽPSymbol…-…ð ‡2 À§…=Ó ‡2 À…-Ó ‡2 ÀD …+Ó ‡2 ÀU…-Ó ‡2 ÀÔ…+Ó„û ÿ‚Œ Arial~…-…ð ‡2 …0} ‡2 é; …2} ‡2 ‹ …1} ‡2 …2} ‡2 :…2}„û€þ‚Œ Arial©…-…ð ‡2 À…1× ‡2 Às …2× ‡2 Àf…1× ‡2 ÀÕ….k †& ÿ…û ‚¼"Systemn…-…ð… 2 …-‡2 ÷ºƒ{…ûýã#ƒÿ,ˆßF—x<Ä< ƒP†=þÿƒ  ÿÿþÿ'þÿþÿþÿÿÿ‚Œ Arial‡-ÿÿÿ¡1(BªRËZªR‰JHB(B(BHBHBBç9e)ã$!ÿ!ÿÿ!ÿ„ymbÿ‚      $‚ ….k ‡2 Àž ….k  ¡P†=ˆ1‚,ƒt‚,ƒt       ƒ1‚ ‡ À` † „û€þ‚¼” ArMathType€…ûýゃl†þÿ ƒ{…ûýゎPSybol…-…ðŽ2 ÷ ¬ÿƒ…-…ð ‡2 ÀÚÑlpoN€h ‚ …‚ÿ‚….ƒ1‚ ‡ †&ÿƒ‰Àÿ¸ÿÀ ¸ “& MathType…ûþ傎PSymbol„…-‡2 óƒ(…ûþ傎PSymbol„…-…ð‡2 ó ƒ)…ûþ傎PSymbol„…-…ð‡2 óeƒ(…ûþ傎PSymbol„…-…ð‡2 óT ƒ)…û+ý䂎PSymbol……-…ð‡2 /*ƒ{…û+ý䂎PSymbol……-…ð‡2 /» ƒ}„û€þ‚Œ Arial©…-…ð ‡2 à†…1× ‡2 ๅ2× ‡2 àÏ…1ׄû ÿ‚Œ Arial……-…ð ‡2 –…2} ‡2 4Ñ …2}„û€þ‚ŽPSymbol…-…ð ‡2 àk…-Ó ‡2 à´…-Ó„û€þ‚Œ Arial……-…ð ‡2 à—…tk ‡2 àÈ…tk ‡2 àà …tk ‡2 àO …tk ‡2 à-…,k ‡2 à» …,k ‡2 àX ….k †& ÿ…û ‚¼"Systemn…-…ð#‚Œ Arial©…-Ään9 k…2× ‡2 Àf…1× ‡2 Õ….k †& ÿ“soft Equation 2.0 ˆDS EqC‡2 :…2}„û€þ„ÿ‚Œ Arial~¢- ˜ˆ1 ˜ƒt˜ˆƒ°ûcƒÿ†bol~…-…ðŠ2 Ó¬ßF—x<Ä< ƒP†=þÿƒ  ÿþÿþ ÿÿþÿþÿþÿþÿþÿ ãIJ0„•­×½¾XÆyΙΙÎyΙÎXÆ÷½Öµÿ‚Œ Arial©‚-ÿÿ…k„û€þ‚ªPSymb1(BªRËZªR‰JHB(B(BHBHBBç9e)ã$!ÿ!ŠPSymbol~…-‡2 Ó½ÿƒ‰Àÿ¹ÿ Y ˆ& Ma…óÎà‚ßF‹à<¼<~…-…ð‡2 Ó©…2} ‡2 :…2}û€þPSymbol~…-‡2 Ó½      ƒk„û€þ‚ŒPSymb ….k ‡2 Àž ….k Œ Arial~…-¨ð P†=ˆ1‚,ƒt‚,ƒt       A<Ak‚ † „û€þ‚¼… Ar±¨lpD€ m‚ …‚ÿ‚….ƒ1‚ ‡ À€†&ÿƒ‰Àÿ­ÿ@m “& MathType€…ûýゎPSymbol‚…-‡2 ö®ƒ{…ûýゎPSymbol‚…-…ð‡2 öCƒ}„û€þ‚Œ Arial©…-…ð ‡2 À…P ‡2 ÀN…tk ‡2 À;…tk„û€þ‚ŽPSymbol…-…ð ‡2 À~…=Ó„û€þ‚Œ Arial©…-…ð ‡2 À=…1ׄû ÿ‚Œ Arial©…-…ð ‡2 ³…2}„û€þ‚Œ Arial©…-…ð ‡2 À¿…,k ‡2 À¬…,k ‡2 Àà…,k †& ÿ…û ‚¼"Systemn…-…ðŠ÷½÷½÷½€þ‚ŽPSymbol„´…-Ó„û€þ‚¦ˆ1†-ƒt–(–)‚,˜ƒt à—…tk ‡2 àȃtk‡2 àà …tk ‡2 àO ƒk ‡2 à-…,k †2 à» …,k ‡2 àX ….k ‚&H?lpÁ@D t‚ …‚ÿ‚….ƒ1‚ ‡ €€†&ÿƒ‰Àÿ®ÿ@. “& MathTypep…ûþ傎PSymbol……-‡2 ³½ƒ(…ûþ傎PSymbol……-…ð‡2 ³½ƒ)…ûæýÛ‚ŽPSymbol……-…ð‡2 ¿Èƒ(…ûæýÛ‚ŽPSymbol……-…ð‡2 ¿ ƒ)…ûæýÛ‚ŽPSymbol……-…ð‡2 ¿½ƒ(…ûæýÛ‚ŽPSymbol……-…ð‡2 ¿ƒ)„û€þ‚¼Œ Arial©…-…ð ‡2  ;…f~ ‡2  ã…bê ‡2  R …bê ‡2  Ž …bê ‡2  8…bê ‡2  e…bê„û€þ‚Œ Arial……-…ð ‡2  I…tk ‡2  É…tk ‡2  Ô…tk„û€þ‚ŽPSymbol …-…ð ‡2  §…=Ó ‡2   …+Ó ‡2  Y …+Ó ‡2  •…+Ó ‡2  ?…+Ó„û ÿ‚Œ Arial……-…ð ‡2 â…0} ‡2 Q …0} ‡2 €…1} ‡2 7…0} ‡2 f…2} ‡2 ôZ…2}„û€þ‚Œ Arial©…-…ð ‡2  °…1× ‡2  L…2× ‡2  ˆ …2× ‡2  õ….k †& ÿ…û ‚¼"Systemn…-…ð€€€&‚ŽPSymbol…-ƒÿ.…ˆ2’ –{–}‚.œôœôœ ÿÿ‚èÿþÿ    þÿƒÿÿÿ“& MathType€…ûýã#‚ÿÿÿÿ¡-/153013"7!%8%8ÿEdFi]}  AEquationÿÿ„-9Xystemn…-…ð…ÿÿ ‡2 Àþ….k Š& ÿÿŽPSymbol…-2 mpObj6ˆystemn…-…ðˆÿÿÿƒ‰Àÿ­ÿ`m ƒ&“uatiPSymbol‚…-†2 ö®!†bol‚…-…ð†2 öCƒ}„û€þ‚Œ Ar ³…2}„û€þ‚ÿˆlp¨ {Ä  1‚ …‚ÿ‚….ƒ1‚ ‡ @ †&ÿƒ‰Àÿ®ÿ`î “& MathTypeP„û€þ‚Œ Arialó…-‡2  !‘Here,×~×k‡2  þ Ÿ the basiskk××k××ÀTÀ‡2  o Ÿ functionsko××ÀkT××À‡2  ä— are 1,k×~×k×k ‡2  Z‡ tkk‡2  Ÿ“ and tk×××kk ‡2  #…,k ‡2  …,k ‡2  ….k ‡2  å…tk„û ÿ‚Œ Arial©…-…ð ‡2 ôi…2} ‡2 ô~…3} †& ÿ…û ‚¼"Systemn…-…ð‡2  ….k ‡2  å‡2 ….i †& ÿ…û ‚¼"Systmn…-…ðÿ‰  É…tk ‡2  Ôƒtk„û€þ‚PSymbol B…+Ó ‡2  •…+Ó ‹2  ð ‡2  I…tk „2  F ‰à/o¹´»’@À`àe)Mk!  2 €…1} ‡2 70 ‡2 f…2} ‡2 ôZ"†al©…-…ð †2  °…1× ‡2  L…2× ‘2 PSymbol……-ˆð€€…f~ ‡2  ã…bê •2 oot Entryž€ÜÿÜ€ÜsÜ€ÜòÿÜ#ƒÿ„û€þ‚¼š ArialhVSÿTea…À¸ uÿ,ea…Àtion Native ƒ ÿƒÀ‚FŠMicr.ŽPSymbol‚ÿþÿ%„û€þ‚Œ Arial…À¼FEquation N¥²”ï{mkmkÎsï{|ï{||®smkMk®s0„ˆ2’ –{–}‚.œôœôœ‡2  É…tk ‡2  Ô=ï=ï=ï=ï=ï=ï=ï=÷^÷^÷^÷^ï=ï=ï=¢{0„²”Ç9e)mkï{ï{ï{ï{ï{ï{ï{ï{ï{…ûæýÛ‚ÃPSymbol-/153013"7!%8%8œôœôœøøøøøôœôœôœôœ„÷½÷½‰Ò ƒ@d‰h ÿÿÿÿûlp##ö¼  l‚ …‚ÿ‚….ƒ1‚ ‡ €à†&ÿƒ‰Àÿ¹ÿ 9 “& MathTypeð„û€þ‚Œ Arial…-‡2  8 ŸThe coeffiî××kÀ××ooT‡2  ß Ÿcient for ÀT××kko×~k‡2  ¦ Ÿthe basis k××k××ÀTÀk‡2   Ÿfunction "o××ÀkT××kŠ ‡2  V‡1"׊ ‡2  ö is kTÀk ‡2 ¤…0× ‡2 ¿¤…0ׄû€þ‚ŽPSymbol…-…ð ‡2 ªï…æ‘ ‡2 Ãï…è‘ ‡2 Õï…ç‘ ‡2 ªp…ö‘ ‡2 Ãp…ø‘ ‡2 Õp…÷‘„û€þ‚Œ Arial©…-…ð ‡2  O….k †& ÿ…û ‚¼"Systemn…-…ð‡2 ªpÿÿÿÿÿƒÀ‚FŒMicrosoft EqF ‰Àø¹´»ÿ‰2 €…1} ‡2 70‘oot Entry/‚ÿ‚ÀF ‰À">»‚@ƒ Ar>‚ ÿ”}  A<A1‰ PIC;ƒÿA<A1‚ ‡META9‰mpObj70 ‡2 f…2} ƒ2 ô    þÿþÿÿþÿÿ«tusWindows=1 ExecuteDecodedFil  ð ‡2  I…tk ÿÿ“2 PSymbol……-‚ð!ÿ!ÿÿ“2 PSymbol……-‚ð›€ÜÿÜ€ÜsÜ€Üòÿ) lpT€ =‚ …‚ÿ‚….ƒ1‚ ‡ € †&ÿƒ‰Àÿ¬ÿÀ , “& MathType0…ûéýゎPSymbol‚…-‡2 ùòƒ(…ûéýゎPSymbol‚…-…ð‡2 ù÷ƒ)…ûéýゎPSymbol‚…-…ð‡2 Âùƒ(…ûéýゎPSymbol‚…-…ð‡2 Âþƒ)„û€þ‚Œ Arial‚…-…ð ‡2 Û@…x¼ ‡2 Û‚…ti ‡2 Ûð…ti ‡2 Û¼…ti ‡2 ÛW …ti ‡2 ¤@…y¶ ‡2 ¤‰…ti ‡2 ¤÷…ti ‡2 ¤„…ti„û€þ‚ŽPSymbol…-…ð ‡2 ÛÕ…=Ó ‡2 Û¸…-Ó ‡2 ÛK …+Ó ‡2 ¤Ü…=Ó ‡2 ¤€…-Ó„û€þ‚Œ Arial‚…-…ð ‡2 Û…6Ö ‡2 Ûà…9Ö ‡2 Û{ …4Ö ‡2 ¤…4Ö ‡2 ¤¨…3Ö„ûàþ‚Œ Arial…-…ð ‡2 38…2  ‡2 3Î …3  ‡2 ün…3  ‡2 ü …2 „û€þ‚Œ Arial‚…-…ð ‡2 Ûu …,i ‡2 ¤§ ….i †& ÿ…û ‚¼"Systemn…-…ðƒ ¿ˆ1†+ƒtˆ1†+ƒt ˆ2R6DURP†îýPhšAónV†îýP‹Fø‹V…Àc†À‚ø…Àd½ebbie Johnson0\\WORDPUSH\MARKET¥3À^_ÉÊ jjjšfrÏ‹ø ÿtçÇFæ$‡ÀVÀ‡2 à‹ Š FunctionsLÿ„ ÿÿŸ FunctionsiéÖÖÀiVÖÖÀ‚hÿþÿ   þÿþÿþÿþÿÿÿ„  ÿ„þÿÿÿÿÿ‡2 ൅.i ‡2 àŽÿAÿkko×~k‡2  ¦ „the „ÿÿ‹"Systemn…-…ðj>–)˜‚.½‰÷½ooT2 ×ooT‡2  ß cient fo1‚ ‡ €à%MathTypeð„û€þ‚ƒÿ›The coeffiî××kÀ××ooŸ‰/ "1" is asis k××k××ÀTÀk‡2  function "o××ÀkT× £function "o××ÀkT×"Systemn…-…ð“€Ô9ÿÿâ0ÿÿÌ1ÿÿþ2‚ŽPSymbolƒ-bÿ„ܶ߂üÿ„ܶ߂ƒÿ Ÿnction "1" ‡2  O….k †& ÿˆ2 ªpŸ\)^'_,e3D}=OŠ2Ex":^'>X/E'C‚ ÿ‰mpObj;‰mpObjŠhTypeð„û€þ‚ObjInfo‚…ø‘ ‡2 Õp…÷‘% †& ÿ…û ‰mpObjÿÿþ ÿ]Tlp“ É@n þ‚ …‚ÿ‚….ƒ1‚ ‡ @€ †&ÿƒ‰Àÿ­ÿ@ í “& MathType …ûéýゎPSymbol‚…-‡2 ºöƒ(…ûéýゎPSymbol‚…-…ð‡2 ºƒ)…ûéýゎPSymbol‚…-…ð‡2 ƒÿƒ(…ûéýゎPSymbol‚…-…ð‡2 ƒ ƒ)„û€þ‚Œ Arial‚…-…ð ‡2 œ@…x¼ ‡2 œ†…ti ‡2 œ©…ti ‡2 e@…y¶ ‡2 e…ti ‡2 eo…ti ‡2 eš…ti„û€þ‚ŽPSymbol…-…ð ‡2 œæ…=Ó ‡2 œF…-Ó ‡2 eï…=Ó ‡2 e7…-Ó„û€þ‚Œ Arial‚…-…ð ‡2 œ(…2Ö ‡2 œk…2Ö ‡2 e1…2Ö ‡2 e\…2Ö„ûàþ‚Œ Arial…-…ð ‡2 ½' …2 „û€þ‚Œ Arial‚…-…ð ‡2 œ …,i ‡2 eØ ….i †& ÿ…û ‚¼"Systemn…-…ð‚‚…-…ð †2 àŠ‡  ƒ‚ …û ‚¼"Systmn…-…ðƒ ÿ‘à2@_ÿÿà€˜1ü…Àc†À‚ø…Àd½ebbie Johnson0\\WORDPUSH\MARKET¥3À^_ÉÊ jjjšfrÏ‹ø ÿtçÇFæ$‡ÀVÀ‡2 à‹ Š FunctionsLÿ„ ÿ" FunctionsiéÖÖÀiVÖÖÀ‚hÿþÿ   þÿþÿþÿþÿÿÿÿ!ÿ…ð ‡2 ÛÕ…=Ó ‚2 ‡2 ¤€…-Ó„û€þ‚ ÿÿÿÿ!Î(–)†=ˆ4ƒt ˆ3 †-¦1ŠRŠRŠRŠRŠRŠRŠRŠRŠRŠRŠRŠRŠRŠRkko×~k‡2  ¦ „the !Š"Systemn…-…ðjŽPSymbol‚…-…ðƒ)„û€þ‚Š Ar ‚‚…-…ð ‡2 Û‚ ÿ‡  `à‡€à ‰‚$!ÿ›The coeffiî××kÀ××ooŸˆ1†+ƒtˆ1†+ƒt ŽPSymbol…-ÿ!‚…2  ‡2 3Î …3  oÀ‚h„û€þ‚Œ Arial‚"„ ‡2 ÛÕ…=Ó 2 Û¸-…ð ‡2 Ûƒ6Ö2 ×ooT‡2  ß „cien" ‡2 ¤„…ti„û€þŸnction "1" B‡2 Âþƒ)„û€þ ÿÿˆlp’ÀÄ Ý‚ …‚ÿ‚….ƒ1‚ ‡   †&ÿƒ‰Àÿ¤ÿà D “& MathTypeð„û€þ‚Œ Arial…- ‡2 À-…aÖ ‡2 Àö…ti ‡2 À€…ti„ûàþ‚Œ Arialn…-…ð ‡2 ý…5  ‡2 p…5  ‡2 þ…3 „û€þ‚Œ Arial…-…ð ‡2 À¦…6Ö ‡2 À…,i ‡2 À¡…,i ‡2 §Ü…5Ö ‡2 çà…3Ö ‡2 À‡t,ii ‡2 §í ‡2tÖi ‡2 çL ‹-2t€Öi ‡2 À} ….i„ûàþ‚Œ Arialn…-…ð ‡2 ?# …2 „û€þ‚ŽPSymbol…-…ð ‡2 ¿*…æ“ ‡2 î*…è“ ‡2 ê*…ç“ ‡2 ¿¦…ö“ ‡2 ø“ ‡2 ꦅ÷“ ‡2 ¿ …æ“ ‡2 î …è“ ‡2 ê …ç“ ‡2 ¿Ú …ö“ ‡2 îÚ …ø“ ‡2 êÚ …÷“ †& ÿ…û ‚¼"Systemn…-…ðƒ¦ …û ‚¼"Sytemn…-…ð‚‚…-…ð †2 àŠ‡  ƒ‚ …û ‚¼"Systmn…-…ðƒ ÿ‘à2@_ÿÿà€˜1ü-…ûéýã‚­PSymbol‚¥3À^_ÉÊ jjjšfrÏ‹ø ÿtçÇFæ$‡ÀVÀ‡2 à‹ Š Functions þÿþÿ…Ó ‡2 eï…=Ó ‡2 e7/‚h†bol‚…-…ð…2 ƒÿ   þÿþÿþÿþÿÿ!†bol‚…-…ð†2 ƒ !ÿ„V ‡2 ¤€…-Ó„û€þ‚ ÿ„ ‚.z|c„} ÿƒ‚…-…ð ‡2 œ@$¢(–)†=ˆ4ƒt ˆ3 †-„û€þ‚ˆ Arialkko×~k‡2  ¦ ‡the …ûéý゘PSymbol…Àudž ‡é>=ê†ui‹†H ‚ ÿ ŽPSymbol‚…-…ð…-Ó„û€þ‚ƒ ÿ‚Ÿebbie Johnson0\\WORDPUSH\MA ÿ …û ‚¼“"t Entry›The coeffiî××kÀ××oo‚…2  ‡2 3Î ƒ3 ƒ¼ ‡2 œ†…ti †2 œ©KBlp21žJ v‚ …‚ÿ‚….ƒ1‚ ‡ ` ,†&ÿƒ‰Àÿ®ÿ`, “& MathType`…ûþ傎PSymbol……-‡2 ³þƒ(…ûþ傎PSymbol……-…ð‡2 ³þƒ)„û€þ‚Œ Arial……-…ð ‡2  ,…p× ‡2  Š…tk ‡2  …a× ‡2  &…a× ‡2  ¿…tk ‡2  Ÿ …a× ‡2  l …tk ‡2  f…a× ‡2  &…tk„û ÿ‚Œ Arial……-…ð ‡2 N…n} ‡2 ô¢…n}„û€þ‚ŽPSymbol…-…ð ‡2  â…=Ó ‡2  …+Ó ‡2   …+Ó ‡2  ë …+Ó ‡2  F…+Ó„û ÿ‚Œ Arial……-…ð ‡2  …0} ‡2 …1} ‡2  …2} ‡2 ôð …2}„û€þ‚Œ Arial…-…ð ‡2  Þ….k ‡2  L….k ‡2  º….k ‡2  6…,k ‡2  ,….k‡2  Å Ÿ where thek××~×kk×× ‡2  Ç‡ akׇ2  f Ÿ are scalak×~×kÀÀ×Tׇ2  ê# Ÿrs or vect~Àk×~k¼×Àk ‡2  *‹ors×~À„û ÿ‚Œ Arial……-…ð ‡2 …i1 †& ÿ…û ‚¼"Systemn…-…ð™ÿÿÿÿaB¾MkÇÿ‚ÿ‚….ƒ1"†ÿ‘oot Entry‡2 p…5  ‡2 þ3#‚ ÿ$‚Èjÿƒs„ss„壅ñ£‚þÿÿÿÿÿƒ‚ …‚ÿ„ÿÿÿÿ› à` À€ € `ÿÿ”soft Equation 2.0 ‰DS Equati!…).‚… ArJA<AÝ‚ “’&H Ý‚ …‚ÿ‚….‚™TXT.lnkREADME~1.LNK7MathTypeð„û€þ‚$…5  ‡2 p…5  ‡2 À¦…6Ö ‡2 À, Arialn…-…ð DŒ Arial…-†ð <ùlpÒ¸  ç‚ …‚ÿ‚….ƒ1‚ ‡ ` †&ÿƒ‰Àÿ¥ÿ` “& MathTypeà…ûþ傎PSymbol‚…-‡2 ³Óƒ(…ûþ傎PSymbol‚…-…ð‡2 ³Óƒ) …úƒ"…-‡=‡=Ù „û€þ‚Œ Arial©…-…ð ‡2  &…r~ ‡2  _…tk ‡2 ©…tk ‡2 ©í …tk ‡2 ©– …tk ‡2 ËÐ …tk„û€þ‚ŽPSymbol…-…ð ‡2  ·…=Ó ‡2 ©ß…-Ó ‡2 ©Æ…+Ó ‡2 ©k …+Ó ‡2 Ë©…-Ó„û€þ‚Œ Arial©…-…ð ‡2 ©ü…1× ‡2 ©…2× ‡2 ©ê…3× ‡2 ËÆ…1× ‡2 ËË…2ׄû ÿ‚Œ Arial…-…ð ‡2 ýq …3} ‡2 ý …4} ‡2 T …2}„û€þ‚Œ Arial©…-…ð ‡2  þ ….k †& ÿ…û ‚¼"Systemn…-…ð†  Ç‡ akׇ2  f Ÿ ae scalak×~×kÀÀ×T׆  ê# •rs or vect~Àk×~k‡×Àk ‡2  *‰ors×~Àƒ× ‡2  Ç‡ ak׆2  ÿ‚….ƒ1‚ …û ‚¼"Systemn…-…ð“ÿÿÿÿaBð™ÿÿÿÿaB¾MkÇÿk ‡2  *‹ors×~À„û ÿ‚¤ Arial…ation Native¿‰2 p…5  ‡2 þ3#‚ ÿ$‚È‚*‹ors×~À„û ÿƒÿÿÿþÿÿ‹PSymbol……-‡2 ³þÿÿÿ…ÿ‡   …+Ó ‡2  ë ƒ+Ó‡2  ê# “rs or vect~Àk×ÿk¼×Àk ‡2  *‡ors×ÿÿA…).‚… Ar ‡2  º….k ‡2  6¡,a ƒn ˜ƒt ƒn‰jInfo“’&H ݆ Œ Arial……-£ are scalak×~×kÀÀ×T×78 ÿpglpÁ”  R‚ …‚ÿ‚….ƒ1‚ ‡ €†&ÿƒ‰Àÿ©ÿÀ) “& MathTypep…ûéýゎPSymbol‚…-‡2 ¾ƒ(…ûéýゎPSymbol‚…-…ð‡2 ¾Òƒ)„û€þ‚¼Œ Arial§…-…ð ‡2  $…li ‡2  Ú…bé„û€þ‚Œ Arial§…-…ð ‡2  …0Ö„ûàþ‚Œ Arial§…-…ð ‡2 Ç…0 „û€þ‚ŽPSymbol…-…ð ‡2  °…=Ó„û€þ‚Œ Arial§…-…ð ‡2  m….i †& ÿ…û ‚¼"Systemn…-…ðÿ…û ‚¼‚…1Ö ‡2 Ã…2Öÿÿ‰2 "ë …4  ‡2 i 2„û€þ‚Œ Arial…-…ð ‡2 Àß ƒi †& ÿƒû‚¼"Systemnƒ-…ð‚Š Arialÿ„ˆÐÏࡱႼ"SystemnŒ-ation Equation.2ô9²q ˆÐÏࡱá>‹orsÖ€À„ûàþ‚Œ Arial¨…-ˆð þ‚Œ Arial¨†- ÿ( …PIC;ƒÿ$L„Ù „û€þ‚‡ Aria!ÿ‚èjÿ!ÿÿÿÿ……ð ‡2  &‡r~ ÿÿ!ÿ‡2 ©Æ…+Ó ‡2 ©k ÿÿ„û ÿ‚‹ Arial!ÿÿÿEŒ Arial©…-ƒð“Òni ç‚ …‚ÿ‚…." ÿMathTypeà…ûþ傊hTypeà…ûþ傃f‹O ËÆ…1× ‡2 Ë˃2׃) …úƒ"‚-‰Bð“ÿÿÿÿaB¾Mk …úƒ"…-„ ‡2  &…r~ †2  _$….k †& ÿ‰€lp3ÁÆ  R‚ …‚ÿ‚….ƒ1‚ ‡ € †&ÿƒ‰Àÿ©ÿ`) “& MathTypep…ûéýゎPSymbol‚…-‡2 ¾ƒ(…ûéýゎPSymbol‚…-…ð‡2 ¾…ƒ)„û€þ‚¼Œ Arial§…-…ð ‡2  $…li ‡2  ™…bé„û€þ‚Œ Arial§…-…ð ‡2  î…1Ö„ûàþ‚Œ Arial§…-…ð ‡2 ‚…1 „û€þ‚ŽPSymbol…-…ð ‡2  i…=Ó„û€þ‚Œ Arial§…-…ð ‡2   ….i †& ÿ…û ‚¼"Systemn…-…ð…ûéýれPSymbolÿAÿ8‚ƒŸ4ÿ„xÿ‰>þÿ ÿ‚ÀŒFEquati fÿ‡ObjE„Ìž>‚ˆÑñ†•º…Ole;‚ ÿ( …PIC8†ÿ$L …ð ‡2  $…li „2  !ÿ„2 ‚Èuation.2ô9²qÿ• ˆ0 ‚. ÿÿ!ÿÿÿŠ@ À @À@… @ Œ@à P(Pÿ™EdFiÙY  AEquatiÿ`‰ ¶G#4u4X‚˜ `‰ öH#4u4X… ÿƒ‚…-…ð‡2 ¾Ò-ˆÐÏࡱáÿ‚è(j“Á~  R‚ …‚ÿ‚¦.ΊŸÌ‰ŸÈŒ¡Ç£Ç‘¥È‘§Ë‰žÄ‹ŸÈ’¥Ë¢µÛ‚Œ Arial§‚-(ƒÿ1ÿƒ)„û€þ‚¼… Ar%ÿ‚þ‚Œ Arial§‰-ð ‡2  …0Ö¤ûàþœôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœ¿¶lp“ h2  Ÿ‚ …‚ÿ‚….ƒ1‚ ‡ € †&ÿƒ‰Àÿ¥ÿ@ ¥ “& MathTypeÀ…ûéýゎPSymbol~…-‡2 žƒ(…ûéýゎPSymbol~…-…ð‡2 žÝƒ) …úƒ"…-‡ ‡Ì „û€þ‚¼Œ Arial~…-…ð ‡2 €$…li ‡2 ……bé ‡2 …5…bé„û€þ‚Œ Arial~…-…ð ‡2 €…0Ö ‡2 €…5Ö ‡2 « …2Ö„ûàþ‚Œ Arial~…-…ð ‡2 å …0  ‡2 å …1 „û€þ‚Œ Arial~…-…ð ‡2 €º….i ‡2 €ï ….i„û€þ‚ŽPSymbol…-…ð ‡2 €Á…=Ó ‡2 …!…+Ó †& ÿ…û ‚¼"Systemn…-…ðƒ=ÿÿÿÿÿ‚ŽPSymbol‚…-‡2 Ó„û€þ‚‡ Aria‰>þÿ  ‚ÿ‚ÀF,ÿ‡Equa#‚ ÿ(ÿ‡PIC;ƒÿ$L ‡META9‚ ÿ$‚ÈŒ Arial§…-…ð Dÿ„g2™Tï=0ï= Arialÿƒi fÿ…OÿÿMathTypep…ûéýã‚ÿÿ‡-…ð…ûéýãÿÿ#œóœóœóœ’”ï{Žs‡ `@ 3Á„†èè\ "Systemn…-…ð…ûéýã‚“PSymbol& ÿ…û ƒ1‚ ‡ € "Systemn…-‘ðMathTypep…ûéýゎPSymbol‚…-…2 ¾/ˆÐÏࡱá‚ ÿ…ð ‡2 ‚…1 ‚û…li ‡2  ™…bé…ûi fÿƒO…ð ‡2  î…1Öÿ™EdFiÙY  AEqua…ð ‡2   ….i „û€þ‚‘PSymbol"„û€þ‚‹ Arial§>5lp 0 ˆ‚ …‚ÿ‚….ƒ1‚ ‡ À@ †&ÿƒ‰Àÿ§ÿ g “& MathTypep…ûéýゎPSymbol……-‡2 þσ(…ûéýゎPSymbol……-…ð‡2 þÛƒ)„û€þ‚¼Œ Arial……-…ð ‡2 à@…f€ ‡2 à…aÖ ‡2 àÊ…bé ‡2 àÜ …cÖ„û€þ‚Œ Arial……-…ð ‡2 à_…ti ‡2 à…ti ‡2 àí…ti„û€þ‚ŽPSymbol…-…ð ‡2 à¿…=Ó ‡2 à·…+Ó ‡2 à¶ …+Ó„ûàþ‚Œ Arial……-…ð ‡2 8ž…2 „û€þ‚Œ Arial…-…ð ‡2 à …,i †& ÿ…û ‚¼"Systemn…-…ð‚3†4„ '…9'…Ë*…¾3ÿ¿í½ÿ¿î¾ÿ¿ï¾ÿ¿ð¿ÿ¾ñÀÿ¾=ï=ï=ï=ï=ï=ï=ï=ï=ï=ï=ï=ï=ï=ï=ï=^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½ÿœôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½œôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœc-c,c-c-c-c,c-c-c-c,c-c-c-c,c-ÿc½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½c-c,c-c-c-c,c-c-c-c,c-c-c-c,c-cc-c-c-c,c-c-c-c,c-c-c-c,c-c-c-cc,cMk,cMc,cMk,cMc,cMk,cMc,cMkÿ,c½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½c-c-c-c,c-c-c-c,c-c-c-c,c-c-c-cc,cMk,cMc,cMk,cMc,cMk,cMc,cM…k,c,i ‡°õò ‡ÿÿÿ ÿ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½œôœôœôœôœôœôœôœôœôœôœôœôœôœôœôœ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½ ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½b‚¼Œ Arial~…-%i fÿ…Oÿ!„l…-…ð ‰2 €Áÿÿ‡-…ð…ûéýãƒ)Ñ`‚g2‘¼ï=˜ï= ÿa ÿŒ Arial§…-ƒð\ ‰jInfo…ûéýã‚‹PSymbol‚È#"Systemn…-ƒðÿÿPSymbol‚…-†2 ¾ZQlpêöÀh f‚ …‚ÿ‚….ƒ1‚ ‡ €@†&ÿƒ‰Àÿ¹ÿ9 “& MathTypeð…ûþ傎PSymbol‚…-‡2 ³½ƒ(…ûþ傎PSymbol‚…-…ð‡2 ³½ƒ)„û€þ‚¼Œ Arial‚…-…ð ‡2  ;…f~„û€þ‚Œ Arial‚…-…ð ‡2  I…tk ‡2  Ä…tk ‡2  ‹ …tk„û€þ‚ŽPSymbol…-…ð ‡2  ›…=Ó ‡2 ªÒ…æ‘ ‡2 ÃÒ…è‘ ‡2 ÕÒ…ç‘ ‡2 ªÿ…ö‘ ‡2 Ãÿ…ø‘ ‡2 Õÿ…÷‘ ‡2  A…+Ó ‡2 % …-Ó ‡2 ªd…æ‘ ‡2 Ãd…è‘ ‡2 Õd…ç‘ ‡2 ªÆ …ö‘ ‡2 ÃÆ …ø‘ ‡2 ÕÆ …÷‘ ‡2  M …+Ó ‡2 ªp …æ‘ ‡2 Ãp …è‘ ‡2 Õp …ç‘ ‡2 ªñ…ö‘ ‡2 Ãñ…ø‘ ‡2 Õñ…÷‘„û€þ‚Œ Arial‚…-…ð ‡2 p…1× ‡2 ¿p…1× ‡2 ø …2× ‡2 ¿‰ …0× ‡2 8…1× ‡2 ¿%…0ׄû ÿ‚Œ Arial…-…ð ‡2 ôF…2}„û€þ‚Œ Arial‚…-…ð ‡2  š….k †& ÿ…û ‚¼"Systemn…-…ð‚Œ Arial …PIC;ƒÿÿ‰META8Ÿ½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½÷½„ÿ$ÿ¿ ƒÿÿ¡=ï=^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^÷^ÿÿquation Nativeÿƒ…û ‚¼‹"Systemÿ¡NÛS‡áKÙDxÒKÙM‚ÜJÙLÛSˆâSˆâLÿÿ…2 „û€þ‚£ ƒt ˆ2 †+bƒt†+c* ¨¸…ð ‡2 à_…ti E“ .H ˆ¥ ZyVRoRNkTQk_\vNLiZVsA?\][xMKib`‚,…ÿƒ‰Àÿ§ÿ g ƒ&‡ ……-…ð ‡2 à@ƒ(…ûéýゎPSybol……-…ð†2 þÛi fÿ$…f€ ‡2 à…aÖ ‡2 Ê…bé ‡2 àÜ …cÖ„ u l lpIöÀž %‚ …‚ÿ‚….ƒ1‚ ‡ €À†&ÿƒ‰Àÿ¹ÿ€9 “& MathTypeð…ûþ傎PSymbol‚…-‡2 ³½ƒ(…ûþ傎PSymbol‚…-…ð‡2 ³½ƒ)…ûþ傎PSymbol‚…-…ð‡2 ³;ƒ(…ûþ傎PSymbol‚…-…ð‡2 ³* ƒ)…ûþ傎PSymbol‚…-…ð‡2 ³¿ƒ(…ûþ傎PSymbol‚…-…ð‡2 ³®ƒ)„û€þ‚¼Œ Arial‚…-…ð ‡2  ;…f~„û€þ‚Œ Arial‚…-…ð ‡2  I…tk ‡2  ¶…tk ‡2  "…tk ‡2  :…tk ‡2  …tk„û€þ‚ŽPSymbol…-…ð ‡2  ›…=Ó ‡2 ªÒ…æ‘ ‡2 ÃÒ…è‘ ‡2 ÕÒ…ç‘ ‡2 ªS…ö‘ ‡2 ÃS…ø‘ ‡2 ÕS…÷‘ ‡2  Š…-Ó ‡2  ° …+Ó ‡2 ªÓ …æ‘ ‡2 ÃÓ …è‘ ‡2 ÕÓ …ç‘ ‡2 ªT …ö‘ ‡2 ÃT …ø‘ ‡2 ÕT …÷‘ ‡2  …-Ó ‡2  y…+Ó ‡2 ªœ…æ‘ ‡2 Ü…è‘ ‡2 Õœ…ç‘ ‡2 ª…ö‘ ‡2 Ã…ø‘ ‡2 Õ…÷‘„û€þ‚Œ Arial‚…-…ð ‡2 š…1× ‡2 ¿‡…0× ‡2  ¥…1× ‡2 ˆ …0× ‡2 ¿ˆ …0× ‡2  C…2× ‡2  )…1× ‡2 Q…0× ‡2 ¿d…1ׄû ÿ‚Œ Arial…-…ð ‡2 ɵ …2} ‡2 ô–…2}„û€þ‚Œ Arial‚…-…ð ‡2  -….k †& ÿ…û ‚¼"Systemn…-…ð‡2 Q…0× …2 ¿ÿþÿþ ÿÿÿÿÿ„ ÿÿ‹ Arial‚…-…ð ‚2 …$ÿÿ‡bol‚…-‡2 ³½ƒ(„p …æ‘ ‡2 Ãp …è‘ êö( ІèèŽPSymbol…-…ð ‚è…ÿƒ‰Àÿ¹ÿ9 …&ÿ!…ÿƒ‰Àÿ¹ÿ9 ƒ&MathTypeð…ûþå‚$‡2 8…1× ‡2 ¿%‰0bol‚…-…ð†2 ³½ƒ)„û€þ‚¼‰ Ar½÷½„ÿ…f~„û€þ‚Œ rial‚…-…ð ‹2   ªd…æ‘ ‡2 Ãd…è‘ÿˆ1ˆ0Š–(–)‚.‡2  ›…=Ó ‡2 ªÒ…æ…-…ð ‡2 ôFÿƒ…ø‘ ‡2 Õÿ…÷‘ †  A…+Ó ‡2 % …-Óÿ¿ ƒAÀ‚W´w9w9 …ö‘ ‡2 ÃÆ …ø‘ ‡2 Æ …÷‘ ‡2  M …+Ó !ÍÄlpè+€N k‚ …‚ÿ‚….ƒ1‚ ‡ à€†&ÿƒ‰Àÿ¸ÿ@˜ “& MathTypep…ûéýゎPSymbol……-‡2 σ(…ûéýゎPSymbol……-…ð‡2 Ûƒ)…ûéýゎPSymbol……-…ð‡2 Óƒ(…ûéýゎPSymbol……-…ð‡2 ̃)…ûéýゎPSymbol……-…ð‡2 ܃(…ûéýゎPSymbol……-…ð‡2 Õƒ)„û€þ‚¼Œ Arial……-…ð ‡2 @…f€ ‡2 î…bé ‡2 … …bé ‡2 ±…bé„û€þ‚Œ Arial……-…ð ‡2 _…ti ‡2 P…ti ‡2 @…ti ‡2 Y…ti ‡2 ±…ti„û€þ‚ŽPSymbol…-…ð ‡2 ¿…=Ó ‡2 !…-Ó ‡2 r …+Ó ‡2 *…-Ó ‡2 ž…+Ó„ûàþ‚Œ Arial……-…ð ‡2 `ç…0  ‡2 'Y …2  ‡2 `p …1  ‡2 `±…2  ‡2 X>…2 „û€þ‚Œ Arial…-…ð ‡2 =…1Ö ‡2 : …2Ö ‡2 F…1Ö ‡2 ï….i †& ÿ…û ‚¼"Systemn…-…ð…Ô@v‚Œ Arial……-„ ‡2 a…1Ö †2 5…0Ö ‡2  …1Ö ‡2 Û …0Ö ‡2 é …0Ö ‡2 a…2Ö ‡2 m1 ‡2 ^…0Ö ‡2 Y‡2 ^…0Ö ‡2 Y1„û€þ‚Œ Arial‚…-…ð ‡2  -ƒ.k†& ÿ…û ‚¼"Systemn…-‡2 ªS…ö‘ ‡2 ÃSƒøÿþÿþ ÿÿ‚ŽPSymbol…-‡ð ‡2 š…1× †2 ¿‡!ÿÿ‹ Arial…-…ð ‚2 …ÿƒ‰Àÿ¹ÿ€9 ‰& Maÿÿ‚h MathTypeð…ûþ傆þÿ ÿƒÀ‚F®ÿ¨Öÿ§Õþ©Õþ©Õþ©ÕþªÖÿ¬Öÿ°Ûÿ±Üÿ®ÙÿPSymbol…-…ð ‚¼"Systemn‚-# ÿ5…ÿƒ‰Àÿ¹ÿ9 ƒ&C‡2 8…1× ‡2 ¿%0ƒCƒ)„û€þ‚¼Š ArT …÷‘ ‡2  …-Ó …f~„û€þ‚ƒ Aÿˆ1ˆ0Š–(–)‚.‡2 ªÓ …æ‘ ‡2 ÃÓ  ‡2 Ü…è‘ ‡2 Õœÿƒ…ø‘ ‡2 Õÿ…÷‘ „û€þ‚¼‰ ArialÉÀlpô ƒ€F +‚ …‚ÿ‚….ƒ1‚ ‡ À †&ÿƒ‰Àÿ½ÿ€ ½ “& MathType…ûéýゎPSymbol~…-‡2 þσ(…ûéýゎPSymbol~…-…ð‡2 þÛƒ)…ûéýゎPSymbol~…-…ð‡2 þµ ƒ(…ûéýゎPSymbol~…-…ð‡2 þÁ ƒ)„û€þ‚¼Œ Arial~…-…ð ‡2 à@…f€ ‡2 à.…bé„û€þ‚Œ Arial~…-…ð ‡2 à_…ti ‡2 àÅ…B ‡2 àE …ti„ûàþ‚Œ Arial~…-…ð ‡2 @'…i@ ‡2 @Ñ…i@ ‡2 8Ñ…n  ‡2 ° …i@ ‡2 "š…n „û€þ‚ŽPSymbol…-…ð ‡2 à¿…=Ó„ûàþ‚ŽPSymbol…-…ð ‡2 °d…=ž„û€ý‚ŽPSymbol…-…ð ‡2 K…åÊ„ûàþ‚Œ Arial…-…ð ‡2 ° …0 „û€þ‚Œ Arial…-…ð ‡2 à" …,i †& ÿ…û ‚¼"Systemn…-…ðƒ „2 °…2Ö ‡2 F…1Ö ‡2 ï….i †& ÿ …û ‚¼Œ"Systemn…-…ð…Ô@v‚Œ Arial……-‡META‚À‚F‘Microsoft EqÛ …0Ö ‡2 é …0Ö ‡2 a…2Ö ‡2 m1 ‡2 ^…0Ö ‡2 Y‡2 ^…0Ö ‡2 Yƒ1ÿ‡-…ð ‡2  -ƒ.k†& ÿ…û „… …bé ‡2 ±…bé‡2 =…1Ö ‡2 :  ™Equation Natÿ‚ŽPSymbol…-‰ðûéýゎPSymbol…"‚……-…ð‡2 Õÿ‹ Arial…-…ð ‚2 "ÿƒ‰Àÿ¹ÿ€9 & Mabol……-…ð†2 ̃)„û€þ‚¼¥ ArHdšCb™Lk¨No®TtµWzºV|½V}ÁV€ÃVÄ‚……-…ð‡2 ܃Eq…2  ‡2 X>…2 “û€þPSymbol…-…ð ‚¼"Systemn‚-ÿÿ ÿ6…ð ‡2 _…ti %…ûéýã‚”PSymbol2 8…1× ‡2 ¿%0…ð ‡2 =…1Ö ‚2 ÿ„T …÷‘ ‡2  …-Ó …f~„û€þ‚‡ 5…0Ö ‡2  …1Ö …ûéýれPSymbolÿˆ1ˆ0Š–(–)‚.#š‘lp€è "‚ …‚ÿ‚….ƒ1‚ ‡   †&ÿƒ‰Àÿ¤ÿ`D “& MathTypeð…ûéýゎPSymbol……-‡2 Þêƒ(…ûéýゎPSymbol……-…ð‡2 Þöƒ)…ûéýゎPSymbol……-…ð‡2 Þöƒ(…ûéýゎPSymbol……-…ð‡2 Þï ƒ)„û€þ‚Œ Arial……-…ð ‡2 À#…B ‡2 Àz…ti ‡2 §È…nÖ ‡2 ç…iV ‡2 À±…ti ‡2 Às …ti„ûàþ‚Œ Arial……-…ð ‡2 /…i@ ‡2 /…n  ‡2 7…i@ ‡2 çu …n  ‡2 ç¾ …i@„û€þ‚ŽPSymbol…-…ð ‡2 ÀÚ…=Ó ‡2 ¿…æ“ ‡2 î…è“ ‡2 ê…ç“ ‡2 ¿‡…ö“ ‡2 ø“ ‡2 ꇅ÷“ ‡2 ÀD …-Ó„ûàþ‚ŽPSymbol…-…ð ‡2 ç …-ž„û€þ‚Œ Arial…-…ð ‡2 À` …1Ö ‡2 À ….i †& ÿ…û ‚¼"Systemn…-…ð‹0Ž;ƒn0†Ž;ÿ‰àd¹|…º„Àÿ‡Ole;‚ ÿÿ‰­ ï….i †& ÿ'ƒÿ$L ‡META8ÿ‚hÿþÿ…-…ð ‡2 @'‡i…ðƒ „2 ° ™Equation Natÿ)ÿ‚Œ Arial~‡-ÿÿ˜icrosoft Equation 2.0 †DS Eqÿ!œs b��@ñÿ��Normal���‚ ��‡ �À �ƒ&�� ÿY�‚�À�‚F�“Microsof��ð��ûéýã�‚�’��PSymbol��ð�� �‡2 à_�…t�i� �„2 à��…��‚ÿ‚��….��ƒ1��‚ ��‡ �À �ÿ��MathType���…ûéýã�‚&�ƒ(��…ûéýã�‚�‡��PSy��‚ ��‡ �À �ƒ&��„û€þ�‚¼�� Arial��bol�~�…-���…ð��†2 þµ �ƒ(��…ûéýã�‚�‡��PSy�‚�À�‚F�‰Microsof� ÿ�†al��~�…-���…ð� �…2 à@�…f�€� �‡2 à.�…b�é��†û�n� �Equation.2�ô9²q �½��´���lp����ó3ö�. ���������� ��$�����‚ 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\€Ð�€��0œª€�‚ã‚¡™€�‰€�‚€�‚€�㤓h€�‰€�‚ÿ��To learn what this book is about, ��click here.������To go to the Table of Contents, ��click here.����T���#���ü���þ��1���o ��ÿÿÿÿÿÿÿÿ���þ��@��~ ��Introduction to the CAGD LaboratoryB������ª��@��(��� €4�€��>�˜˜B¤ª€ �‚ÿ�What This Book Is About��ñ���¿���þ��1��2��� 2€�€��2�˜œª€ �‚€�€ �€�‚‚ÿ�Welcome!��Welcome to �Interactive Curves and Surfaces�, where computer-aided geometric design (CAGD) comes to life with interactive, user-paced tuition.�With this electronic book, you will�� ��Ô��@��Q��L��� f€©�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ€ �€�‚ÿ�·��become familiar with standard ways of creating curves and surfaces, including Bézier curves, B-splines, and parametric surface patches,��·��understand the mathematical tools behind the generation of these objects, and the development of computer-based CAGD algorithms,��·��explore the behavior and characteristics of the most popular CAGD curves with interactive test benches,��·��understand the uses of 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The Interactive Curves and Surfaces tutorial behaves the same:��p��%����D��K��� d€M�€��r˜ìª:‚l€ �€�ƒâcÆ6=€�‰€�†"€���‚€ �€�ƒ‚ÿ�·��Use the left mouse button to click on ��highlighted words.�� The left button does almost everything. The mouse pointer turns into a hand � when it's over something you can usefully click on.��·��If you're running Windows 95, use the right mouse button to bring up a menu of useful tasks:��6������Ô��z��2��� 4€ �€��r ˜ìª:‚l€�†"€���‚ÿ����™���q���D�� ��(��� €â�€��2˜ìª‚l€�‚ÿ�From this menu, you may copy text to the clipboard, search for a topic, print a topic, or do some other tasks.����Ñ��z��, ��H��� ^€§�€��r˜ìª:‚l€ �€�ƒ†"€���†"€���‚€ �€�ƒ‚ÿ�·��The buttons along the top edge of the screen are standard on-line help controls, but notice the exit button �, which exits the book, and the browse buttons �. These provide a linear study flow through the book, presenting material in a logical sequence of steps.��·��The interactive applications are started by clicking their graphical symbol. Once running, the applications are controlled separately from the book. Each has its own controls and instructions.��|���O��� ��¨ ��-��� *€ž�€��r˜ìª:‚l€ �€�ƒ‚ÿ�·��The references to the Bibliography may be found in the accompanying text.��(������, ��Ð ��%��� €�€��0�¦¨€�‚ÿ���2������¨ �� ��.��� ,€ �€��0¦¨€�†"€���‚ÿ����*������Ð ��, ��&��� €�€��0�¦¨€�‚‚ÿ����R���!��� ��~ ��1��� 2€B�€��pî¨:‚l㤓h€�‰€�‚ÿ��Go to the Table of Contents����1�������, ��¯ ��1��� ��ÿÿÿÿÿÿÿÿ���¯ ��ÿÿÿÿˆ ��<������~ ��ë ��(��� €(�€��>�˜˜B˜ª€ �‚ÿ�Highlighted Words�����s���¯ ��ˆ ��*��� $€æ�€��8�B˜¨€�€�‚ÿ�Highlighted words may either display a popup window (like this one)�or take you to another section of the book.���B������ë ��Ê ��1���& ��ÿÿÿÿÿÿÿÿ���Ê ����H��Table of Contents<������ˆ ����(��� €(�€��>�˜˜B˜ª€ �‚ÿ�Table of Contents��(������Ê ��.��%��� €�€��0�œª€�‚ÿ���Ø���|�������\���#ˆ€ø�âT&òñSè��2€�€�� �ªã‚¡™€�‰€�‚€�‚ÿ�&€>€�� �ª€�€ �€�‚‚ÿÿÿ��What This Book Is About�������An introduction to �Interactive Curves and Surfaces�, and how to use the electronic book.���”���>���.��š��V���#|€|�âT&òñSè��2€�€�� �ªã4Ô¬ö€�‰€�‚€�‚ÿ�€2€�� �ª€�‚‚ÿÿÿ��About the Authors�������Short biographies of the authors.��� ��²�����°@��X���#~€e�âT&òñSè��:€�€�� �ªã’Û@ú€�€�€�‰€�‚‚ÿ�€L€�� �ª‚‚ÿÿÿ��Topic 1:���Iš��°@��ˆ ��ntroduction to CAGD�����An introduction to the basics of CAGD. This topic discusses background and the type of material needed to understand curves and surfaces.���Ñ���z���š��A��W���#~€ô�âT&òñSè��:€�€�� �ªã‹¼ŸÞ€�€�€�‰€�‚‚ÿ�€R€�� �ª‚‚ÿÿÿ��Topic 2:���Preliminary Mathematics�����A summary of the mathematical tools needed to understand the rest of the book.���·���`���°@��8B��W���#~€À�âT&òñSè��:€�€�� �ªã=Ë4h€�€�€�‰€�‚‚ÿ�€D€�� �ª‚‚ÿÿÿ��Topic 3:���The Bézier Curve�����The behavior and characteristics of this fundamental curve.���á���‰���A��C��X���#~€�âT&òñSè��:€�€�� �ªã 8ûU€�€�€�‰€�‚‚ÿ�€>€�� �ª‚‚ÿÿÿ��Topic 4:���Interpolation�����Interpolating curves are designed to run through a set of existing points. Common types are introduced.���Á���j���8B��ÚC��W���#~€Ô�âT&òñSè��:€�€�� �ªãS‹FZ€�€�€�‰€�‚‚ÿ�€4€�� �ª‚‚ÿÿÿ��Topic 5:���Blossoms�����Blossoms provide an intuitive method for creating Bézier and B-spline curves.���Ô���}���C��®D��W���#~€ú�âT&òñSè��:€�€�� �ªã!}¤÷€�€�€�‰€�‚‚ÿ�€H€�� �ª‚‚ÿÿÿ��Topic 6:���The B-Spline Curve�����The B-spline curve is a very popular curve in industry today and is covered in detail.���ÿ���§���ÚC��­E��X���#~€O�âT&òñSè��:€�€�� �ªã#jõ×€�€�€�‰€�‚‚ÿ�€B€�� �ª‚‚ÿÿÿ��Topic 7:���Rational Curves�����Rational curves introduce the idea of associating weights with the control points. This permits greater variety in the curve forms.���Ã���l���®D��pF��W���#~€Ø�âT&òñSè��:€�€�� �ªã*/Aµ€�€�€�‰€�‚‚ÿ�€4€�� �ª‚‚ÿÿÿ��Topic 8:���Surfaces�����This topic extends CAGD from two to three dimensions, creating curved surfaces.���´���]���­E��$G��W���#~€º�âT&òñSè��:€�€�� �ªãôì%€�€�€�‰€�‚‚ÿ�€R€�� �ª‚‚ÿÿÿ��Topic 9:���Images and Applications�����A gallery of applications and instructive images.���¿���h���pF��ãG��W���#~€Ð�âT&òñSè��:€�€�� �ªã¨ü™¥€�€�€�‰€�‚‚ÿ�€X€�� �ª‚‚ÿÿÿ��Topic 10:���Other Curves and Surfaces�����This topic introduces additional curve and surface types.���$������$G��H��"��� €�€����€�ÿ��E������ãG��LH��1�����ÿÿÿÿz����LH��˜H�� Á��Introduction to CAGDL���!���H��˜H��+��� &€B�€��>�˜˜Bœœ€ �€�‚ÿ�Topic 1: Introduction to CAGD���F��� ���LH��ÞH��&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn��ª���m���˜H��ˆI��=��� J€Ú�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��what CAGD is about,��·��some of the history of CAGD,��·��about some typical applications of CAGD tools.��(������ÞH��°I��%��� €�€��0�œª€�‚ÿ���2������ˆI��âI��.��� ,€ �€��0¦¨€�†"€���‚ÿ����*������°I�� J��&��� €�€��0�œª€�‚‚ÿ����G������âI��SJ��,��� (€6�€��pîª:‚l€ �‚€�‚ÿ�What CAGD Is All About����²���Œ��� J��K��&��� €�€��0�¤ª€�‚ÿ�Computer-aided geometric design (CAGD) is a new field that initially developed to bring the advantages of computers to industries such as��(������SJ��-K��%��� €�€��0�œª€�‚ÿ���k���/���K��˜K��<��� H€^�€��4˜¦¨‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��automotive��·��aerospace��·��shipbuilding��(������-K��ÀK��%��� €�€��0�¤ª€�‚ÿ���…��L��˜K��EN��9��� @€›�€��0�œª€�€ �€�‚‚‡"€���‚ÿ�CAGD expanded rapidly and now pervades many areas, from pharmaceutical design to animation. We are surrounded by products that were first visualized on a computer. These products were modified and refined entirely within the computer; when the product entered production, the tools and dies were produced directly from the geometry stored in the computer. This process is known as �virtual prototyping�.���Computer visualizations of new products reduce the design cycle by easing the process of design modification and tool production.��© 1993 Autodesk, Inc. Reprinted with permission.��ã��»��ÀK��4€��(��� €w�€��0�œª€�‚‚‚ÿ��CAGD is based on the creation of curves and surfaces and is accurately described as curve and surface modeling. Using CAGD tools with elaborate user interfaces, designers create and refine their ideas to produce complex results. They combine large numbers of curve and surface segments to realize their ideas. However, the individual segments they use are relatively simple, and it is at this level thEN��4€��H��at the study of CAGD is concentrated.���2������EN��f€��.��� ,€ �€��0¦¨€�†"€���‚ÿ����*������4€��€��&��� €�€��0�œª€�‚‚ÿ����O���(���f€��߀��'��� €P�€��p�‘€ª€ �‚ÿ�A Design Challenge: The Need for CAGD��(������€����%��� €�€��0�œª€�‚ÿ�����ô���߀��!‚��&��� €é�€���œ€�‚‚ÿ�When creating products and artwork, designers face tasks such as this: You are given two points in a plane and two directions associated with the points. Find the curve that passes through the points that is tangent to the given directions.���1��������R‚��-��� *€ �€��œ€�†"€���‚ÿ����Ä���›���!‚��ƒ��)��� €7�€���œ€�‚‚‚‚‚ÿ��This is a simple pencil-and-paper task for anyone who is familiar with parametric forms, the types of curves used in CAGD.��CAGD tools are meant to be���;������R‚��Qƒ��,��� (€�€��4˜¦¨‚l€ �€�ƒ‚ÿ�·��intuitive��>������ƒ��ƒ��+��� &€&�€��0¦¨‚l€ �€�ƒ‚ÿ�·��simple to use��‡��_��Qƒ��…��(��� €¿�€���œ€�‚‚‚‚ÿ��Sometimes the mathematics underlying the tools becomes quite sophisticated, yet the result is meant to be easily understood and geometrically intuitive.��The technical person often benefits from these intuitive and visually related tools when considering deeper mathematical problems. The geometry of CAGD is very amenable to visual demonstration.��(������ƒ��>…��%��� €�€��0�œª€�‚ÿ���2������…��p…��.��� ,€ �€��0¦¨€�†"€���‚ÿ������ã���>…��„†��1��� 0€Ç�€��0�œª€�‚‚€ �‚€�‚‚‚ÿ����CAGD: From Points to Teapots���There is a natural progression of the geometry behind CAGD. With small incremental steps, it is possible to describe complex objects in terms of simple primitives, such as points and lines.���[���)���p…��߆��2��� 4€R�€��pìª2‚l€ �€�ƒ€�€�‚ÿ�·���Control Points: The Start of CAGD���D��ø���„†��#ˆ��L��� f€ó�€��0�œª€�‚€�‡"€���âNúbl€�‰€�âµûß/€�‰€�‚‚ÿ����Take four points in a plane and connect them to form a polygon. The points are known as ��control points��, and the polygon as the ��control polygon.�� The control points and polygon determine the approximate shape of the curve to be formed.���[���)���߆��~ˆ��2��� 4€R�€��pìª2‚l€ �€�ƒ€�€�‚ÿ�·���A CAGD Favorite: The Bézier Curve���ý���½���#ˆ��{‰��@��� N€}�€��0�œª€�‚€�‡"€�� �âKhè€�‰€�‚‚ÿ����A special curve known as the ��Bézier curve�� may be generated by the control points. Note that while the curve passes through the endpoints, it only comes close to the other points.���v���G���~ˆ��ñ‰��/��� .€Ž�€��pìª2‚l€ �€�ƒ€�‚ÿ�·���Three-Dimensional Control Polygons for Surfaces (Control Meshes)��ï���¯���{‰��àŠ��@��� N€a�€��0�œª€�‚€�‡"€�� �âGë%3€�‰€�‚‚ÿ����Bézier curves behave just as well in three dimensions as in two. This figure shows the control polygons for a three-dimensional object consisting of ��Bézier patches.�����[���)���ñ‰��;‹��2��� 4€R�€��pìª2‚l€ �€�ƒ€�€�‚ÿ�·���Three-Dimensional Wireframe Model���¬���y���àŠ��ç‹��3��� 6€ô�€��0�œª€�‚€�‡"€�� �‚‚ÿ����When the Bézier curves are created and connected in three dimensions, a wireframe model of the object is produced.���Y���'���;‹��@Œ��2��� 4€N�€��pìª2‚l€ �€�ƒ€�€�‚ÿ�·���Three-Dimensional Shaded Object���5����ç‹��u��3��� 4€�€��0�œª€�‚€�‡"€�� �‚ÿ���� If the surface produced by the three-dimensional Bézier patches is illuminated and shaded, an object with a realistic appearance results. The Utah teapot, on display at the Boston Museum of Computer History, is a classic in CAGD and computer graphics.��(������@Œ����%��� €�€��0�œ¨€�‚ÿ���2������u��Ï��.��� ,€ �€��0¦¨€�†"€���‚ÿ����I��������Ž��.��� ,€6�€��0�œª€�‚‚€ �‚€�‚ÿ����The History of CAGD����|���K���Ï��”Ž��1��� 2€–�€��0�œ¨€�ãpšŽb€�‰€�‚ÿ�CAGD has a short but interesting history. To learn more, ��click here.����(������Ž��¼Ž��%��� €�€��0�¤¨€�‚ÿ���2������”Ž��îŽ��.��� ,€ �€��0¤¨€�†"€���‚ÿ����(��õ���¼Ž��"À��3��� 4€ë�€���¤€�‚‚€ �€�‚€�‚‚‚ÿ����What Has Been Accomplished in This Topic����The ideas and principles behind CAGD have been introduced, together with the classic example of surface patch use, the Utah teapot. The short but significant history of CAGD hîŽ��"À��H��as been summarized.���2������îŽ��TÀ��.��� ,€ �€��0¦¨€�†"€���‚ÿ����*������"À��~À��&��� €�€��0œ¨€�‚‚ÿ����Ž���S���TÀ�� Á��;��� F€¦�€��0�¤¨ã¤“h€�‰€�‚㋼ŸÞ€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: Preliminary Mathematics����D������~À��PÁ��1���‘��ÿÿÿÿÿÿÿÿ���PÁ��‘Á��H�The History of CAGDA������ Á��‘Á��*��� $€.�€���˜˜B˜€ �€�‚ÿ�The History of CAGD���-����PÁ��¾Ã��'��� € �€���˜˜¦€�‚ÿ�Computer-aided geometric design has mathematical roots that stretch back to Euclid and Descartes. Its practical application began with automated machinery to compute, draft, and manufacture objects with free-form surfaces. Production pressures in the aircraft industry during World War II stimulated many new devices to enhance and accelerate design and manufacturing. For example, in 1944, Liming designed fuselage spars with a "superelliptic" method that could be implemented with an electromechanical calculator.��º��“��‘Á��xÇ��'��� €'�€���˜¦€�‚‚ÿ�Shipbuilders also became interested in CAGD early on for many reasons. One example of their motivation may sound trivial but was a serious impediment to ship design. The only place large enough to draw full-scale plans for a ship was in the loft of the shipbuilders' dry dock. The huge drawings would warp and shrink in the moist air, causing very real manufacturing problems.�Computers provided the greatest stimulus because of their power to enable new ideas. In 1963, Ferguson developed one of the first surface patch systems by which individual curvilinear patches are joined smoothly to create the surface "quilt." He also introduced the notion of parametrically defined surfaces, which has become the standard because it provides freedom from an arbitrarily fixed coordinate system. Vertical tangent vectors can be defined by differentiation, for instance, which is not possible in explicit Cartesian form.����Ú��¾Ã��yË��'��� €µ�€���˜¦€�‚‚ÿ�In the mid 1960s, automotive companies became involved in CAGD as a way to drive milling machines. Car bodies were designed by artists using clay models. Painstaking measurements produced data that could drive numerically controlled milling machines to produce stamp molds. The initial use of CAGD was to represent the data as a smooth surface for numerical control. It soon became apparent that the surfaces could be used for the design.�In 1971, Pierre Bézier reformulated Ferguson's ideas so that a draftsman without any extensive mathematical training could design a surface. Bezier's system, UNISURF, was used by Renault and became a milestone in the development of CAGD. It epitomized the difference between surface fitting and surface design. The purpose of design was to provide the draftsman, who had strong intuition about shape but limited mathematical training, with computer tools that empowered him or her to use the sophisticated mathematics of surface representation.��ò��Ë��xÇ��kÎ��'��� €—�€���˜¦€�‚‚ÿ�In the meantime, the mathematical underpinnings of CAGD continued to advance. De Casteljau examined triangular patches and developed evaluation techniques. Coons [Coons64] unified much of the previous work into a general scheme that became the basis of the early modeler PDGS made by Ford. At General Motors in 1974, Gordon and Riesenfeld exploited the properties of B-spline curves and surfaces for design.�Driven primarily by the automotive, shipbuilding, and aerospace industries, both the mathematics of CAGD and the designer interface tools continued to improve through the 1970s. The first CAGD conference was organized by Barnhill and Riesenfeld in 1974, where the term "CAGD" was first used [Barnhill74].��z��7��yË��ñ�C��� T€o�€���œ€�âVXÛ€�‰€�€ �€�âæ‹19€�‰€�‚ÿ�In the 1980s, the power and versatility of computer aided designing seemed suddenly to be discovered by anyone who had a free-form geometric surface application. Industrial designers were smitten with the power of computer design, and many commercial modelers became the basis of several substantial applications, including CATIA, EUCLIDkÎ��ñ� Á��, STRIM, ANVIL, and GEOMOD. Geoscience used CAGD methods to represent seismic horizons; computer graphics designers modeled their objects with surfaces, as did molecule designers for pharmaceuticals. Architects discovered CAGD, word processing and drafting programs based their interface protocols on free-form curves (��PostScript��), and even moviemakers discovered the power of animating with such surfaces, beginning with �TRON�, continuing through ��Jurassic Park,�� and beyond.��W���&���kÎ��H�1��� 2€L�€���œ€�‚ã*ðþ�€�‰€�‚ÿ����Return to Introduction to CAGD����>��� ���ñ�†�1�����ÿÿÿÿÿÿÿÿ���†�ÿÿÿÿJ�Jurassic Park;������H�Á�*��� $€"�€���˜˜B˜€ �€�‚ÿ�Jurassic Park���‰���a���†�J�(��� €Â�€���¦€ �€�‚ÿ�Jurassic Park� made extensive use of CAGD and�computer graphics to visualize animated objects.��;��� ���Á�…�1���I��ÿÿÿÿÿÿÿÿ���…�ÿÿÿÿ“�PostScript8������J�½�*��� $€�€���˜˜B˜€ �€�‚ÿ�PostScript���Ö���®���…�“�(��� €]�€���¦€�‚ÿ�PostScript is a proprietary page description language�used by typesetters to define elements of printed text,�including letter outlines, text layout, and graphical�images.��?������½�Ò�1��� ��ÿÿÿÿÿÿÿÿ���Ò�ÿÿÿÿœ�Control Points=������“��+��� &€$�€��>�˜˜B˜ª€ �€�‚ÿ�Control Points������g���Ò�œ�&��� €Î�€��0�œª€�‚ÿ�Control points are points in two or more dimensions�that define the behavior of the resulting curve.��A�������Ý�1���º��ÿÿÿÿÿÿÿÿ ���Ý�ÿÿÿÿV�Control Polygons>������œ��+��� &€&�€��>�˜˜B˜ª€ �€�‚ÿ�Control Polygon���;����Ý�V�+��� $€!�€��0�œª€�‚ÿ�The control polygon is formed by connecting the�control points in the correct order. The control�polygon provides a crude analogy of the refined�curve. Note that the control polygon is typically�open (the ends are not coincident), and it may�self-intersect arbitrarily.��?�������•�1���3��ÿÿÿÿÿÿÿÿ ���•�ÿÿÿÿ‰�Bézier Patches=������V�Ò�+��� &€$�€��>�˜˜B˜ª€ �€�‚ÿ�Bézier Patches���·������•�‰�(��� €�€��0�œª€�‚ÿ�A Bézier patch is a three-dimensional extension�of a Bézier curve. It is formed by extruding a�Bézier curve through space to form a surface.��A������Ò�Ê�1���~��ÿÿÿÿÿÿÿÿ ���Ê�ÿÿÿÿ �The Bézier Curve?������‰� �+��� &€(�€��>�˜˜B˜ª€ �€�‚ÿ�The Bézier Curve���þ���Õ���Ê� �)��� €«�€��0�œª€�‚ÿ�The Bézier curve, named after the French researcher�Pierre Bézier, is a simple and useful CAGD curve. It is a�very well behaved curve with useful properties, as you�will discover in Topic 3, "The Bézier Curve."��H������ �O �1���ìV��$„��äƒ� ���O � �=O�Preliminary MathematicsN���$��� � �*��� $€H�€���˜˜B¤€ �€�‚ÿ�Topic 2: Preliminary Mathematics���F��� ���O �ã �&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn��†��@�� �i �F��� Z€�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��the basic math needed for the rest of the book,��·��an overview of parametric forms, a convenient way of describing curves and surfaces,��·��the idea of continuity, to ensure that curves and surfaces join together smoothly,��·��an illustration of linear interpolation, one of the most fundamental concepts in CAGD.��(������ã �‘ �%��� €�€��0�œª€�‚ÿ���2������i �à �.��� ,€ �€��0¦¨€�†"€���‚ÿ����*������‘ �í �&��� €�€��0�œª€�‚‚ÿ����G������à �4 �+��� &€8�€��0¤ª‚l€ �‚€�‚ÿ�The Mathematics of CAGD����|��T��í �°�(��� €©�€���¤€�‚‚‚‚ÿ�CAGD treats points, lines, and surfaces as mathematical objects. That is, they may be described geometrically in two- or three-dimensional space.��This book develops the mathematics in a step-by-step fashion and does not demand a rigorous mathematical background. It is recommended that the reader be familiar with the following topics:���›���_���4 �K�<��� H€¾�€��2˜¤ª‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��geometry of points, lines, and planes��·��equations in parametric form��·��basic calculus��'������°�r�$��� €�€���¤€�‚ÿ���2������K�¤�.��� ,€ �€��0¤¨€�†"€���‚ÿ����*������r�Î�&��� €�€��0�¤¨€�‚‚ÿ����:������¤�@�'��� €&�€��0¤ª‚l€ �‚ÿ�ParametricÎ�@� � Forms��,��÷���Î�@A�5��� 8€ï�€��0�¤¨€�‚ã#±V€�‰€�‚‚‚ÿ��CAGD relies on parametric forms to describe curves and surfaces. Many students of CAGD do not initially appreciate the subtlety and importance of this form. If you are confident with parametric forms then ��click here�� to skip this section.����>������@�~A�'��� €.�€��0¤ª‚l€ �‚ÿ�The Parametric Curve��Œ��R��@A� C�:��� B€§�€��0�¤¨€�‚‚‚‡"€�� �€ �€�‚‚‚ÿ��Typically, when a student takes mathematics, a curve is presented as a graph of a function f(x).���As x is varied, y = f(x) is computed by the function f, and the pair of coordinates (x, y) sweeps out the curve. This is called the �explicit� form of the curve.��From a design standpoint the explicit form is deficient in several ways.��?������~A�IC�,��� (€&�€��2˜¤ª‚l€ �€�ƒ‚ÿ�·��Single-Valued��¶���†��� C�ÿC�0��� .€�€��2�˜¤ª€�‡"€���‚ÿ��The curve is single-valued along any line parallel to the y axis. For example, only parts of the circle may be defined explicitly.��@������IC�?D�,��� (€(�€��2˜¤ª‚l€ �€�ƒ‚ÿ�·��Infinite Slope��Þ���®���ÿC�E�0��� .€_�€��2�˜¤ª€�‡"€���‚ÿ��An explicit curve cannot have infinite slope; the derivative f'(x) is not defined parallel to the y axis. Hence there are two points on the circle that cannot be defined.��I������?D�fE�,��� (€:�€��2˜¤ª‚l€ �€�ƒ‚ÿ�·��Transformation Problems��”���n���E�úE�&��� €Ü�€��2�˜¤ª€�‚ÿ�Any transformation, such as rotation or shear, may cause an explicit curve to violate the two points above.��1��ü���fE�+G�5��� 8€ù�€��0�¤¨€�‚â†Lëï€�‰€�‚‚‚ÿ��The parametric form of a curve is not subject to these limitations. Moreover, it provides a method, known as ��parameterization,�� that defines motion on the curve. Motion on the curve refers to the way that the point (x, y) traces out the curve.����G��� ���úE�rG�'��� €@�€��0¤ª‚l€ �‚ÿ�Defining the Parametric Curve��}��U��+G�ïH�(��� €«�€��0�¤¨€�‚‚‚ÿ��A parametric curve that lies in a plane is defined by two functions, x(t) and y(t), which use the independent parameter t. x(t) and y(t) are coordinate functions, since their values represent the coordinates of points on the curve. As t varies, the coordinates (x(t), y(t)) sweep out the curve. As an example consider the two functions:���Q���&���rG�@I�+��� &€L�€��p¤¨È‚¡…€�ƒ‚ÿ�x(t) = sin(t), y(t) = cos(t).�(2.1)��R��Â��ïH�’K���� �€��0�¤¨€�‚€ �€�‚‚È(�ExecProgram("demo1.exe Circle_Demo", 0)�€�‡"€���‰€�‚‚â9€�‰€�âùàé€�‰€�€ �€�€ �€�‚ÿ��As t varies from zero to 2�p�, a circle is swept out by (x(t), y(t)).�������Click on this interactive demonstration to observe the way that the parameter t creates a circle from the two coordinate functions.��CAGD deals primarily with ��polynomial�� or ��rational functions��, not trigonometric functions as shown in the examples above. For example, the circle can also be given by allowing t to vary from -�¥� to +�¥� in the following functions:��(������@I�ºK�%��� €�€��0�¤¨€�‚ÿ���>��� ���’K�øK�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.2)��­��v��ºK�¥N�7��� <€í�€��0�¤¨€�‚‚‚‚‚âÀÙ°i€�‰€�‚ÿ��As an exercise, verify for yourself that the functions in equation 2.2 do indeed generate a circle. Plot points (x(t), y(t)) or write a program to do this for you.��Both equation 2.1 and equation 2.2 yield circles, so how do they differ? It is the parameterization. The motion of the point (x(t), y(t)) is different, even if the paths (the circles) are the same.��A good physical model for parametric curves is that of a ��moving particle.�� The parameter t represents time. At any time t the position of the particle is (x(t), y(t)). Two paths (curves) may be identical even though the motion (parameterization) is different.��x��7��øK�)�A��� P€o�€��0�¤¨€�‚â¨á0΀�‰€�â8~7€�‰€�‚‚‚ÿ��Parametric curves are not constrained to be single-valued along any line (recall the single-valued deficiency of the explicit form), and the slope of a parametric curve segment may be defined vertically. The slope is given by the ��tangent line�� at any point, computed by finding¥N�)� � the ��derivative vector�� (x'(t), y'(t)) at any point t. This vector determines the speed at which the point traces out the curve as t changes.��Curves defined by points whose speed may drop to zero do cause problems that will be considered later under the discussion of continuity.��r���K���¥N�›�'��� €–�€��0�¤¨€�‚‚‚ÿ��Consider the parametric curve given by these two coordinate functions:���>��� ���)�Ù�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.3)��k���E���›�D‚�&��� €Š�€��0�¤¨€�‚‚ÿ��In the next interactive demonstration, note the following points:��²��l��Ù�öƒ�F��� Z€Ù�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��The parameter t moves the point (x(t), y(t)) along the path of the curve.��·��The point's speed varies as t varies. The speed is higher at the ends of the curve.��·��The derivative vector changes in length, reflecting the variation in the speed of the point.��·��In the demonstration, the curve crosses itself, which can easily happen with parametric curves.��(������D‚�„�%��� €�€��0�¤¨€�‚ÿ���Ë���h���öƒ�é„�c��� –€Ò�€��2˜¤ª‚lÈ+�ExecProgram("demo1.exe Tangent_Bezier", 0)�€�‡"€���‰€�‚ÿ�����Click on this interactive demonstration to explore the behavior of the tangent to a Bézier curve.��U���.���„�>…�'��� €\�€��0�¤¨€�‚‚‚ÿ��A convenient notation for equation 2.3 is���>��� ���é„�|…�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.4)����K��>…�ý†�6��� :€—�€��0�¤¨€�‚âÕ2[Ê€�‰€�‚‚‚‚ÿ��Equations 2.3 and 2.4 are the same. We simply save on notation by writing the ��basis functions�� only once, which are then multiplied by the appropriate vectors. When a vector is multiplied by a scalar, each coordinate in the vector is individually multiplied by the scalar.��In general, a parametric polynomial is written as���>��� ���|…�;‡�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.5)��Ž��>��ý†�ɉ�P��� n€}�€��0�¤¨€�‚€�€�€�€�€�€�€�€�€�€�‚‚€�€�€�‚ÿ��where �f�(t) is a vector-valued function, and the �a�'s are vectors. The vectors are not restricted to two dimensions. The �a�'s might be vectors of three dimensions, for instance. In this case the function �f�(t) would have three coordinate functions x(t), y(t), and z(t). The curve would be a curve in space, and the derivative �f�'(t) would be given by the vector of the derivative coordinate functions (x'(t), y'(t), z'(t)).��The general case described by equation 2.5 includes a constant term (�a�0�), which the example given by equations 2.3 and 2.4 does not have.��(������;‡�ñ‰�%��� €�€��0�¤¨€�‚ÿ���2������ɉ�#Š�.��� ,€ �€��0¤¨€�†"€���‚ÿ����,������ñ‰�OŠ�(��� €�€��0¤¨‚l€�‚‚ÿ����@������#Š�Š�'��� €2�€��0¤ª‚l€ �‚ÿ�The Parametric Surface��`��-��OŠ�ï‹�3��� 4€[�€��0�¤¨€�‚€ �€�€ �€�‚ÿ��As with curves, it is typical for the reader to have encountered surfaces explicitly as z = f(x, y). Often called �elevation� surfaces or �terrain�, the height z is given at a point on the plane by computing f(x, y). Such a surface definition shares the same flaws mentioned previously for curves:��â���¤���Š�ÑŒ�>��� J€I�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��They must be single-valued for any point on the plane.��·��They cannot have vertical tangent planes.��·��Transformations may cause the above two difficulties.��"��ø���ï‹�ó�*��� "€ñ�€��0�¤¨€�‚‚‚‚‚ÿ��The parametric form of the surface corrects these problems. In order to define a parametric surface, it is best to first define a parametric curve, and then sweep the curve through space to define the surface.��Consider a planar curve given by���>��� ���ÑŒ�1Ž�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.6)��?������ó�pŽ�'��� €0�€��0�¤¨€�‚‚‚ÿ��Or, in vector form,���>��� ���1Ž�®Ž�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.7)��0��û���pŽ�Þ�5��� 8€ù�€��0�¤¨€�‚‚‚‡"€���‚‚‚‚ÿ��The curve given by x(t) and y(t) looks like this:���Here, the parameter t is limited to the range 0 to 1.��The curve becomes a surface in three dimensions if another parameter s and another coordinate function z are added. Consider, for instance:���>��� ���®Ž�(À�4��� 8€�€��p¤¨È‚Þ�(À� �¡…€�†"€���ƒ‚ÿ���(2.8)��Ò��ž��Þ�úÂ�4��� 6€?�€��0�¤¨€�‚‚‚‡"€���‚‚‚ÿ��When s = 1, the curve defined by equation 2.5 is produced on the plane z = 1. As this curve changes in s, it sweeps out a surface. A parametric surface may be thought of as a bundle of parametric curves; by fixing s or t on a surface, one single curve from this bundle is selected.���In this figure, the planar curve is extruded through the z dimension to become a surface. When s = 0.8, the red curve is produced as t varies between 0 and 1.��In equation 2.8, x(s,t) and y(s,t) have no terms in s, and z(s,t) has no term in t. The terms are limited to simplify the example, but this is not typical. In general the surface may be written as the parametric polynomial��(������(À�"Ã�%��� €�€��0�¤¨€�‚ÿ���>��� ���úÂ�`Ã�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.9)��ž���q���"Ã�þÃ�-��� *€â�€��0�¤¨€�‚€�€�‚‚ÿ��Bold letters indicate vector quantities. The indices of the �a�-vectors correspond to the parametric powers.���2������`Ã�0Ä�.��� ,€ �€��0¤¨€�†"€���‚ÿ����*������þÃ�ZÄ�&��� €�€��0�¤¨€�‚‚ÿ����@������0Ä�šÄ�'��� €2�€��0¤ª‚l€ �‚ÿ�Parametric Derivatives��ã���¯���ZÄ�}Å�4��� 6€_�€��0�¤¨€�‚âÞ³¶€�‰€�‚‚ÿ��The derivative of a parametric curve, in the context of the way in which the curve is parameterized, was mentioned earlier. The derivative function of ��equation 2.3�� is���?��� ���šÄ�¼Å�4��� 8€�€��p¤¨È‚¡…€�†"€���ƒ‚ÿ���(2.10)��U��¿��}Å�É�–��� ú€�€��0�¤¨€�‚âŸ"Ô¦€�‰€�€�€�€�€�‚‚È4�ExecProgram("demo1.exe Simple_Hodograph_Bezier", 0)�€�‡"€���‰€�‚‚€�€�€�€�‚ÿ��The derivative function is itself a parametric curve of degree one less than the original curve. The derivative curve is called the ��hodograph.�� The hodograph of �f�(t) reveals much about �f�(t), especially when viewed graphically.�������Click on this interactive demonstration to watch the way that the hodograph varies with the curve.��Notice in the hodograph demonstration that the derivative vector of �f�(t) is the vector from the origin to �f'�(t) in the hodograph. When the hodograph crosses the x axis, the original curve is parallel to the x axis; when the hodograph crosses the y axis, the curve is parallel to the y axis. The hodograph is discussed further in the section on continuity.�� ��ä��¼Å�Ë�'��� €É�€��0�¤¨€�‚‚ÿ��Recalling that a surface may be considered to be a bundle of curves in both the s and t parameters, it should not be surprising that the derivative of a surface must be given with respect to either the s or t parameter. For example, given a point, the derivative of the surface at that point with respect to the parameter s is simply the derivative of the curve embedded in the surface for a fixed t. At the same point on the surface there is another derivative with respect to t.��:������É�VË�3��� 6€�€��0�¤¨€�‚†"€���‚€�‚ÿ�������3������Ë�‰Ë�/��� .€ �€��p�¤¨È€�†"€���‚ÿ����|��&��VË�Î�V��� z€O�€��0�¤¨€�‚‡"€�� �€�€�‚‚âa²J}€�‰€�€�€�€�€�€�€�‚ÿ���This figure shows the normal (�N�) to a surface, and the two derivatives with respect to the two parameters s and t.��The two derivative vectors define a tangent plane at the point (s, t). The ��cross product�� of these two vectors yields the normal vector �N� to the tangent plane. �N� is also the normal to the surface at the point. This is very useful in applications that render the surface by computing light reflections using the surface normal. Typically the normal vector �N� is made to be unit length by dividing by its length, that is,��(������‰Ë�-Î�%��� €�€��0�¤¨€�‚ÿ���3������Î�`Î�/��� .€ �€��p�¤¨È€�†"€��!�‚ÿ����(������-Î�ˆÎ�%��� €�€��0�¤¨€�‚ÿ���2������`Î�ºÎ�.��� ,€ �€��0¤¨€�†"€���‚ÿ����*������ˆÎ�äÎ�&��� €�€��0�¤¨€�‚‚ÿ����4��� ���ºÎ�Ï�'��� €�€��0¤ª‚l€ �‚ÿ�Continuity��k��$��äÎ��G��� \€I�€��0�¤¨€�‚â7 ò€�‰€�‚‚€ �€�€ �€�€ �€�‚ÿ��The notion of ��continuity ��was developed for explicit functions to describe when a curve does not break or tear. If it meets these conditions, it is describeÏ�� �d as C0. C0 continuity is defined by the popular description, "A curve is continuous if it can be drawn without lifting the pencil from the paper."��If the derivative curve is also continuous, then the curve is �first-order� �differentiable� and is said to be C1 continuous. Extending this idea, it is said that a curve is �Ck differentiable� if the kth derivative curve is continuous.��è��´��Ï�w�4��� 6€k�€��0�¤¨€�‚‚‚‡"€��"�‚‚‚ÿ��Practically, this means that a C1 continuous curve will not kink. Higher degrees of continuity imply a smoother curve.���Two curves are shown here, one that is C0 and one that is C1. The C0 curve has a kink, while the C1 curve is generally smooth.��Unfortunately, continuity does not always result in the expected smoothness when viewed parametrically. The coordinate functions (such as x(t), y(t), and z(t)) may be first-order differentiable and still kink. All that continuity guarantees for parametric curves is that the motion of the particle (the parameterization) is smooth; there are no sudden jumps in velocity. It does not say that the path of the particle (the curve) is smooth.����Ý���|�(��� €»�€��0�¤¨€�‚‚‚ÿ��The traditional notion of C1 continuity does not, in fact, ensure much about the curve's properties. Imagine, for instance, a particle that travels in a straight line but has distinct jumps in velocity. It is not C1, but the curve is certainly smooth. Conversely, it is possible to have a C1 curve with a kink in it. This can occur when the velocity of the particle goes to zero, where it changes direction and starts up again. This is illustrated in the following figure:���~��5��w�ú�I��� `€m�€��0�¤¨€�†"€��#�‚‚â÷1â�€�‰€�â­,€�‰€�‚ÿ����Mathematicians have developed the concept of a ��manifold�� as a new way of describing continuity. In CAGD there is a simpler concept to achieve the same end. It is the idea of ��geometric continuity.�� If a curve is C0, it is G0 continuous. If a curve's tangent direction changes continuously then it is G1 continuous. Its magnitude may jump discontinuously, but the curve is still G1. Hence a particle traveling at erratically changing speeds may still trace out a smooth curve if its direction changes smoothly. This is illustrated in the following figure:��V��#��|�P �3��� 4€I�€��0�¤¨€�‚†"€��$�‚‚‚‚ÿ�����If a C1 curve has kinks because its derivative goes to zero at a point, then this curve will not be G1, since the tangent direction changes discontinuously at the kink. Hence the notion of geometric continuity provides a useful way to understand the smoothness of a curve or surface.���2������ú�‚ �.��� ,€ �€��0¤¨€�†"€���‚ÿ����*������P �¬ �&��� €�€��0�¤¨€�‚‚ÿ����3��� ���‚ �ß �'��� €�€��0¤ª‚l€ �‚ÿ�Exercises��P���(���¬ �/ �(��� €P�€��2˜¤ª‚l€�‚ÿ�Describe physical situations in which����Ä���ß �J �W��� |€‰�€��2˜¤ª„l€�ƒƒâ“u{€�‰€�‚ƒƒâ×¹€�‰€�‚ƒƒâ§~8„€�‰€�‚ÿ�1.�A particle travels with C1 and G1 continuity.���Example.���2.�A particle travels with C1 but not G1 continuity.���Example.���3.�A particle travels with G1 but not C1 continuity.���Example.����(������/ �r �%��� €�€��0�¤¨€�‚ÿ���2������J �¤ �.��� ,€ �€��0¤¨€�†"€���‚ÿ����*������r �Î �&��� €�€��0�¤¨€�‚‚ÿ����>������¤ � �'��� €.�€��0¤ª‚l€ �‚ÿ�Linear Interpolation��Ž���g���Î �š �'��� €Î�€��0�¤¨€�‚‚‚ÿ��Given two points in space, a line can be defined that passes through them both in parametric form:���?��� ��� �Ù �4��� 8€�€��p¤¨È‚¡…€�†"€��%�ƒ‚ÿ���(2.11)��1��� ���š � �'��� €�€��0�¤¨€�‚‚‚ÿ��where���3������Ù �=�/��� .€ �€��p�¤¨È€�†"€��&�‚ÿ����˜���h��� �Õ�0��� 0€Ð�€��0�¤¨€�‚‚‚€�€�‚‚‚ÿ��the two points in space.��Thus �l�(t) is a point somewhere in space, depending on the parameter t.����3��� ���=��'��� €�€��0¤ª‚l€ �‚ÿ�Exercises��(������Õ�0�%��� €�€��0�¤¨€�‚ÿ���ì���ƒ����(@�i���  €�€��4œ¨„lÑ€�ƒ€�€�ƒâÑY¨>€�‰€�‚ƒ€�€�ƒâ >ÿ€�‰€�‚ƒ€�€�ƒâê亀�‰€�‚ÿ�4.�What is the value of �l�(0)?���Answer���5.�What is the value of �l�(1)?���Answer���6.�What is the v0�(@� �alue of �l�(½)?���Answer����Å��U��0�íA�p��� ®€­�€��0�¤¨€�‚È1�ExecProgram("demo1.exe Linear_Interpolation", 0)�€�‡"€��'�‰€�‚‚‚‚‚‚ÿ�������Click on this interactive application to explore the use of linear interpolation between two points on a line.��Linear interpolation is perhaps the most fundamental concept. All subsequent curves and surfaces are defined by repeated linear interpolation in some form.��Other forms of linear interpolation are possible. For example,���3������(@� B�/��� .€ �€��p�¤¨È€�†"€��(�‚ÿ����n���G���íA�ŽB�'��� €Ž�€��0�¤¨€�‚‚‚ÿ��which also gives a straight line through the two points. Note that���3������ B�ÁB�/��� .€ �€��p�¤¨È€�†"€��)�‚ÿ������î���ŽB�×C�(��� €Ý�€��0�¤¨€�‚‚‚ÿ��This is the same straight line (a linear combination of the two points), but it has a different parameterization. That is, the motion of a particle at t is different. In most cases it is preferable to start at t = 0 and end at t = 1.���2������ÁB� D�.��� ,€ �€��0¤¨€�†"€���‚ÿ����*������×C�3D�&��� €�€��0�¤¨€�‚‚ÿ����9������ D�lD�'��� €$�€��0¤ª‚l€ �‚ÿ�Basis Functions��>��þ���3D�ªE�@��� N€ý�€��0�¤¨€�‚â°dG €�‰€�â‰Çúç€�‰€�‚‚ÿ��The final preliminary mathematics that is introduced is that of a polynomial basis. Polynomials such as ��equation 2.5�� are written as a sum of coefficients and simple terms known as ��monomials.�� If the monomials are considered as a collection P,���?��� ���lD�éE�4��� 8€�€��p¤¨È‚¡…€�†"€��*�ƒ‚ÿ���(2.12)�� ��ç��ªE� H�9��� @€Ï�€��0�¤¨€�‚â)Ô<•€�‰€�€ �€�‚ÿ��then the question may be asked whether any polynomial of degree k can be written as a summation of terms, each a product of a coefficient and a basis function from P. If yes, then P is said to span the set of polynomials of degree k. Further, if P is the smallest set to span, then it is a basis for the polynomials of degree k. The set P is a ��basis,�� called a �power basis�. If any monomial is eliminated from P, then not all polynomials of degree k can be written in terms of P.��Ë���£���éE�ÔH�(��� €G�€��0�¤¨€�‚‚‚ÿ��There are other bases. Simple algebra allows us to rewrite parametric functions in another form. For example, consider this parametric polynomial of degree 2:���?��� ��� H�I�4��� 8€�€��p¤¨È‚¡…€�†"€��+�ƒ‚ÿ���(2.13)��e���>���ÔH�xI�'��� €|�€��0�¤¨€�‚‚‚ÿ��This is written with combinations of the following terms:���?��� ���I�·I�4��� 8€�€��p¤¨È‚¡…€�†"€��,�ƒ‚ÿ���(2.14)��f���?���xI�J�'��� €~�€��0�¤¨€�‚‚‚ÿ��This can be rewritten in terms of the following monomials:���?��� ���·I�\J�4��� 8€�€��p¤¨È‚¡…€�†"€��-�ƒ‚ÿ���(2.15)��,������J�ˆJ�&��� € �€��0�¤¨€�‚‚ÿ�as���?��� ���\J�ÇJ�4��� 8€�€��p¤¨È‚¡…€�†"€��.�ƒ‚ÿ���(2.16)��ë���½���ˆJ�²K�.��� *€{�€��0�¤¨€�‚€�€�‚‚ÿ��Try this for yourself. In the example above, the coefficients �b� have a more geometrical and intuitive meaning. This meaning underpins the entire concept of designing with a computer.���2������ÇJ�äK�.��� ,€ �€��0¤¨€�†"€���‚ÿ����.��ú��²K�N�4��� 6€õ�€���¤€�‚‚€ �€�‚€�‚‚‚‚ÿ����What Has Been Accomplished in This Topic����The background to parameterized curves and surfaces has been covered in considerable detail. The hodograph visualized the behavior of the derivative of a curve and clarified the concept of the motion of a point on the curve.��Continuity was introduced to manage the boundaries between multiple curves or surfaces, and extended to include geometric continuity. Finally, linear interpolation was considered as a prerequisite for Bézier curves and blossoming.��'������äK�9N�$��� €�€���¤€�‚ÿ���2������N�kN�.��� ,€ �€��0¦¨€�†"€���‚ÿ����Ò���‚���9N�=O�P��� n€�€��0¤¨‚l€�‚‚㤓h€�‰€�‚ã=Ë4h€�‰€�‚ã’Û@ú€�‰€�‚ÿ�����Go to the Table of Contents�����Go to the next topic: The Bézier Curve�����Go to the previous topic: Introduction to CAGD����A������kN�~O�1���q��ÿÿÿÿÿÿÿÿ ���~O�ÿÿÿÿ{€�Parameterization>������=O�¼O�*��� $€(�€���˜˜B¤€ �€�‚ÿ�Parameterization���³���Œ���~O�{€�'��� €�€���¤€�‚ÿ�Parameterization uses an ind¼O�{€�=O�ependent parameter�or variable to compute points on the curve. It gives�the "motion" of a point on the curve.��A������¼O�¼€�1���r��ÿÿÿÿÿÿÿÿ���¼€�ÿÿÿÿí�Moving Particles=������{€�ù€�*��� $€&�€���˜˜B¤€ �€�‚ÿ�Moving Particle���ô���É���¼€�í�+��� $€“�€���¤€�€�‚ÿ�As the parameter t changes, the coordinate point (x(t), y(t))�traces the curve. This point can be thought of as a particle�that moves under the influence of changes in the value of�the parameter t.���>��� ���ù€�+‚�1���N��ÿÿÿÿÿÿÿÿ���+‚�ÿÿÿÿ;ƒ�Tangent Lines:������í�e‚�*��� $€ �€���˜˜B¤€ �€�‚ÿ�Tangent Line���Ö��� ���+‚�;ƒ�6��� :€C�€���¤€�‚‚†"€��/�‚€�‚ÿ�The tangent line to a curve is the straight line that�gives the curve's slope at a point. This is deduced�from the derivative of the curve at the point.�������B������e‚�}ƒ�1���=��ÿÿÿÿÿÿÿÿ���}ƒ�ÿÿÿÿx„�Derivative Vector?������;ƒ�¼ƒ�*��� $€*�€���˜˜B¤€ �€�‚ÿ�Derivative Vector���¼���†���}ƒ�x„�6��� :€�€���¤€�‚‚†"€��0�‚€�‚ÿ�The derivative vector (x'(t), y'(t)) at the point t is�found by differentiating the functions x(t) and y(t)�with respect to t.�������@������¼ƒ�¸„�1���À���ÿÿÿÿÿÿÿÿ���¸„�ÿÿÿÿ8…�Basis Functions=������x„�õ„�*��� $€&�€���˜˜B¤€ �€�‚ÿ�Basis Functions���C������¸„�8…�;��� F€�€���¤€�†"€��1�‚‚€�†"€��2�‚ÿ��������;��� ���õ„�s…�1���Ù���ÿÿÿÿÿÿÿÿ���s…�ÿÿÿÿ†�Equation 3A������8…�´…�*��� $€.�€���˜˜B¤€ �€�‚ÿ�Recall equation 2.3���'������s…�Û…�$��� €�€���¤€�‚ÿ���6������´…�†�1��� 2€ �€��P�¤È€�†"€��3�€�‚ÿ�����:��� ���Û…�K†�1���t��ÿÿÿÿÿÿÿÿ���K†�ÿÿÿÿ…‡�Hodograph7��� ���†�‚†�*��� $€�€���˜˜B¤€ �€�‚ÿ�Hodograph�����Ì���K†�…‡�7��� <€™�€���¤€�€ �€�€ �€�€�‚ÿ�The word "hodograph" comes from the Greek �hodos��(road, path) and -�graph� (writing, writer). The curve�was first devised by Sir W. R. Hamilton to plot the�direction and velocity of moving particles.���I������‚†�·�1���Ð��ÿÿÿÿÿÿÿÿ���·�ÿÿÿÿU‰�The Vector Cross ProductB������…‡�ˆ�'��� €6�€���˜˜B¤€ �‚ÿ�The Vector Cross Product��E��á���·�U‰�d��� –€Å�€���¤€�‚€�‚€�†"€��4�‚€�‚€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ�The cross product takes two input vectors and produces�a third vector that is perpendicular to the input vectors.���������In the figure, the vector �W� is the cross product of vectors��U� and �V�, that is, �W� = �U� x �V�.��;��� ���ˆ�‰�1���~��ÿÿÿÿÿÿÿÿ���‰�ÿÿÿÿÓŠ�Continuity4��� ���U‰�ĉ�'��� €�€���˜˜B¤€ �‚ÿ�Continuity����ä���‰�ÓŠ�+��� $€É�€���¤€�€�‚ÿ�Continuity implies a notion of smoothness, that is, curves that�are not jagged and that do not break. Commercial applications�of CAGD (for example, car body design) frequently require that�curves and surfaces are continuous.���9������ĉ� ‹�1���û��ÿÿÿÿÿÿÿÿ��� ‹�ÿÿÿÿÎŒ�Manifold6��� ���ÓŠ�B‹�*��� $€�€���˜˜B¤€ �€�‚ÿ�Manifold���Œ��G�� ‹�ÎŒ�E��� X€‘�€���¤€�‚€�‚€�‡"€��5�‚€�‚€�€�‚ÿ�Informally, a manifold is a surface (or a hypersurface in�n dimensions) that is a local deformation of Euclidean�n-space at every point. Hence a 2-manifold looks like a�curved plane locally:���������If a surface is C1 manifold, it implies smoothness, no tearing,�no self-intersecting, and a few other more abstract notions.���E������B‹��1�����ÿÿÿÿÿÿÿÿ����ÿÿÿÿë�Geometric Continuity>������ÎŒ�Q�'��� €.�€���˜˜B¤€ �‚ÿ�Geometric Continuity��š���u����ë�%��� €ê�€���¤€�‚ÿ�Geometric continuity introduces a notation that immediately tells�the designer whether or not the curve is smooth.��=��� ���Q�(Ž�1���Û���ÿÿÿÿÿÿÿÿ���(Ž�ÿÿÿÿÆŽ�Equation (5)A������ë�iŽ�*��� $€.�€���˜˜B¤€ �€�‚ÿ�Recall equation 2.5���'������(Ž�Ž�$��� €�€���¤€�‚ÿ���6������iŽ�ÆŽ�1��� 2€ �€��P�¤È€�†"€��6�€�‚ÿ�����:��� ���Ž���1�����ÿÿÿÿÿÿÿÿ�����ÿÿÿÿ À�Monomials7��� ���ÆŽ�7�*��� $€�€���˜˜B¤€ �€�‚ÿ�Monomials���b���:�����™�(��� €t�€���¤€�‚€�‚ÿ�Monomials are single-termed polynomials. For example,����2������7� À�.��� ,€ �€��P�¤È€�†"€��7�‚ÿ���������������������������������������������������������™� À�ÆŽ�6������™�BÀ�1���s��ÿÿÿÿÿÿÿÿ���BÀ�ÿÿÿÿ{Á�Bases3��� ��� À�uÀ�*��� $€�€���˜˜B¤€ �€�‚ÿ�Basis�����Û���BÀ�{Á�+��� $€·�€���¤€�€�‚ÿ�Given a space S of functions (a collection of functions such�as polynomials, trigonometric, etc.), a set B is a basis if all�functions of S are linear combinations of functions from B,�and B is as small as possible.���<��� ���uÀ�·Á�1���ú���ÿÿÿÿÿÿÿÿ���·Á�ÿÿÿÿuÂ�Polynomials9������{Á�ðÁ�*��� $€�€���˜˜B¤€ �€�‚ÿ�Polynomials���S���+���·Á�CÂ�(��� €V�€���¤€�‚€�‚ÿ�A polynomial is a function of the form����2������ðÁ�uÂ�.��� ,€ �€��P�¤È€�†"€��8�‚ÿ����C������CÂ�¸Â�1���¨��ÿÿÿÿÿÿÿÿ���¸Â�ÿÿÿÿÄ�Rational Functions<������uÂ�ôÂ�'��� €*�€���˜˜B¤€ �‚ÿ�Rational Functions��}���T���¸Â�qÃ�)��� "€¨�€���¦€�‚€�‚ÿ�A rational function is made by dividing one polynomial�by another, for example:����2������ôÂ�£Ã�.��� ,€ �€��P�¦È€�†"€��9�‚ÿ����z���R���qÃ�Ä�(��� €¤�€���¦€�€�‚ÿ�A rational function may contain vector coefficients only�within the numerator.���B������£Ã�_Ä�1���³���ÿÿÿÿÿÿÿÿ���_Ä�ÿÿÿÿÐÄ�The value of l(0)@������Ä�ŸÄ�*��� $€,�€���˜˜B˜€ �€�‚ÿ�The value of l(0):���1������_Ä�ÐÄ�-��� *€ �€���œ€�†"€��:�‚ÿ����B������ŸÄ�Å�1���³���ÿÿÿÿÿÿÿÿ���Å�ÿÿÿÿƒÅ�The Value of l(1)@������ÐÄ�RÅ�*��� $€,�€���˜˜B˜€ �€�‚ÿ�The value of l(1):���1������Å�ƒÅ�-��� *€ �€���œ€�†"€��;�‚ÿ����D������RÅ�ÇÅ�1���µ���ÿÿÿÿÿÿÿÿ���ÇÅ�ÿÿÿÿ8Æ�The Value of l(0.5)<������ƒÅ�Æ�'��� €*�€���˜˜B˜€ �‚ÿ�The value of l(½):��5������ÇÅ�8Æ�0��� 0€ �€���œ€�†"€��<�€�‚ÿ�����]���,���Æ�•Æ�1�����ÿÿÿÿÿÿÿÿ ���•Æ�ÿÿÿÿLÇ�A particle travels with C1 and G1 continuity[���1���8Æ�ðÆ�*��� $€b�€���˜˜B˜€ �€�‚ÿ�A particle travels with C1 and G1 continuity:���\���5���•Æ�LÇ�'��� €j�€���œ€�€�‚ÿ�For example, a car makes a smooth turn on a road.���a���0���ðÆ�­Ç�1���Y��ÿÿÿÿÿÿÿÿ!���­Ç�ÿÿÿÿ¥È�A particle travels with C1 but not G1 continuity_���5���LÇ� È�*��� $€j�€���˜˜B˜€ �€�‚ÿ�A particle travels with C1 but not G1 continuity:���™���q���­Ç�¥È�(��� €â�€���œ€�€�‚ÿ�For example, a car slows to a stop sign, turns its wheels,�and then speeds up again in a different direction.���b���1��� È�É�1���X��ÿÿÿÿÿÿÿÿ"���É�ÿÿÿÿýÉ�A particle travels with G1 but not C1 continuity:_���5���¥È�fÉ�*��� $€j�€���˜˜B˜€ �€�‚ÿ�A particle travels with G1 but not C1 continuity:���—���o���É�ýÉ�(��� €Þ�€���œ€�€�‚ÿ�For example, a baton is passed from one runner to a faster�runner, who stays in the same lane of the track.���A������fÉ�>Ê�1���ûS��z�ÿ �#���>Ê�†Ê�¾ �The Bézier CurveH������ýÉ�†Ê�+��� &€:�€��^�˜˜B˜Œ€ �€�‚ÿ�Topic 3: The Bézier Curve���F��� ���>Ê�ÌÊ�&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn��O����†Ê�Ì�N��� j€�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��what a Bézier curve is,��·��the properties and behavior of the Bézier curve,��·��how to create a Bézier curve,��·��the de Casteljau algorithm for evaluation of a Bézier curve,��·��subdivision, degree elevation, and differentiation of the Bézier curve.��(������ÌÊ�CÌ�%��� €�€��0�œª€�‚ÿ���2������Ì�uÌ�.��� ,€ �€��0¦¨€�†"€���‚ÿ����*������CÌ�ŸÌ�&��� €�€��0�œª€�‚‚ÿ����Q���%���uÌ�ðÌ�,��� (€J�€��pîª:‚l€ �‚€�‚ÿ�Introduction to the Bézier Curve����˜����ŸÌ�ˆÎ�|��� Æ€;�€���œ€�âþqò€�‰€�âé¹M€�‰€�‚‚È)�ExecProgram("demo1.exe Cubic_Bezier", 0)�€�‡"€��=�‰€�‚‚ÿ�The Bézier curve is a good place to start a study of CAGD. It underpins other concepts such as ��B-splines�� and ��surface patches.�� It is visually engaging and exhibits many desirable properties for design.�������Click on the graphic for a demonstration of a cubic Bézier curve.���1������ðÌ�¹Î�-��� *€ �€��œ€�†"€���‚ÿ������Ó���ˆÎ�ÉÏ�=��� H€©�€���œ€�‚‚€ �€�‚‚‚‚†"€��>�‚‚‚‚ÿ����Mathematical Properties of the Bézier Curve���Consider the parabola that passes through (0,1) and (1,0) and is tangent to the x and y axes at these points:�����The parametric form of a parabola looks like���?��� ���¹Î���5��� :€�€��pœß€È‚¡…€�†"€��?�ƒ‚ÿ��ÉÏ���ýÉ��(3.1)��ð���«���ÉÏ��E��� X€W�€���œ€�‚€�€�€�€�€�€�€�€�€�€�‚‚ÿ��where �a�, �b�, and �c� are vector coefficients. �f�(t) is a vector function with two components, that is, �f�(t) = (x(t), y(t)). The above parabola can be written as���?��� �����C�5��� :€�€��pœß€È‚¡…€�†"€��@�ƒ‚ÿ���(3.2)��C�������†�&��� €:�€���œ€�‚‚‚ÿ��This can be rewritten as���?��� ���C�Å�5��� :€�€��pœß€È‚¡…€�†"€��A�ƒ‚ÿ���(3.3)��£��T��†�h�O��� l€©�€���œ€�‚â;ØDa€�‰€�‚‚âNúbl€�‰€�âµûß/€�‰€�‚‚‚‚ÿ��To see the reformation process, ��click here.����This is exactly the same curve as equation 3.2, so what advantage is there in rewriting? The advantage lies in the geometrical meaning of the coefficients: (1,0), (0,0), and (0,1). These are called ��control points��. Together, the control points form the ��control polygon��.��Observe:���o���D���Å�×�+��� &€ˆ�€��˜¦‚l€ �€�ƒ‚ÿ�·��The curve passes through the endpoints of the control polygon.��q���G���h�H�*��� $€Ž�€��œ‚l€ �€�ƒ‚ÿ�·��The curve is cotangent to the control polygon at these endpoints.��û����×�C�k��� ¤€#�€���œ€�‚‚‚È-�ExecProgram("demo1.exe Quadratic_Bezier", 0)�€�‡"€��B�‰€�‚‚‚‚ÿ��The curve in this form is called the Bézier curve, and the observations hold in general for any coefficients. This means that if the coefficients are changed, the curve changes in an easy-to-understand way.�������Now that control points have been introduced, drag the control points in the interactive demonstration to see how the curve changes.��The general form for a quadratic Bézier curve is���?��� ���H�‚�5��� :€�€��pœß€È‚¡…€�†"€��C�ƒ‚ÿ���(3.4)�����¾���C�‚�B��� R€}�€���œ€�‚€�€�€�€�€�€�€�€�€�‚‚ÿ��This is a parabola exactly like equation 3.1, but it is rewritten so that the control points �b�0�, �b�1�, and �b�2� have geometrical significance as the control points of the parabola.���1������‚�³�-��� *€ �€��œ€�†"€���‚ÿ����‰���Z���‚�<�/��� .€´�€���œ€�‚‚€ �€�‚‚‚‚ÿ����Bézier Curves of General Degree���The general form of a Bézier curve of degree n is���?��� ���³�{�5��� :€�€��pœß€È‚¡…€�†"€��D�ƒ‚ÿ���(3.5)��}���N���<�ø�/��� .€œ�€���œ€�‚€�€�€�‚‚ÿ��where �b�i� are vector coefficients, the now-familiar control points, and���=��� ���{�5 �3��� 6€�€��PœÈ‚¡…€�†"€��E�ƒ‚ÿ���(3.6)��Ï��{��ø� �T��� v€ù�€���œ€�‚†"€��F�‚‚â‹à]‹€�‰€�ãí Í€�‰€�‚‚€ �€�‚‚‚‚ÿ�����To learn about the binomial coefficients, ��click here; �� to learn about the Bernstein functions, ��click here.����The collection of Bernstein functions for i = 0, 1, ..., n is the �Bernstein basis.���The Bernstein basis is a key to understanding Bézier curves. Many of the important properties that make Bézier curves useful in design derive from these basis functions.���1������5 �5 �-��� *€ �€��œ€�†"€���‚ÿ����¦���w��� �Û �/��� .€î�€���œ€�‚‚€ �€�‚‚‚‚ÿ����Characteristics of the Bézier Curve���Bézier curves have a number of characteristics that define their behavior.���H������5 �# �,��� (€8�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Endpoint Interpolation��d����Û �‡ �V��� z€�€���ì€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚‚ÿ�The Bézier curve interpolates the first and last points �b�0� and �b�n�. In terms of the interpolation parameter t: �f�(0) = �b�0� and �f�(1) = �b�n�. This property derives from the Bernstein functions, since at the endpoints the Bernstein functions are zero except:���2������# �¹ �.��� ,€ �€���µ€€�†"€��G�‚ÿ����'������‡ �à �$��� €�€���œ€�‚ÿ���D������¹ �$�,��� (€0�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Tangent Conditions��«���…���à �Ï�&��� € �€���ì€�‚‚ÿ�The Bézier curve is tangent to the first and last segments of the control polygon, at the first and last control points. In fact,���2������$��.��� ,€ �€���µ€€�†"€��H�‚ÿ����.������Ï�/�&��� €�€���ì€�‚‚‚ÿ��and���2�������a�.��� ,€ �€��P�ìÈ€�†"€��I�‚ÿ����A������/�¢�'��� €4�€��P�µ€:€�‚‚ÿ��where n is a constant.��'������a�É�$��� €�€���œ€�‚ÿ���=������¢�@�,��� (€"�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Convex É�@�ýÉ�Hull��œ���`���É�®@�<��� H€À�€���ì€�âbè§™€�‰€�€ �€�€ �€�‚ÿ�The Bézier curve is contained in the ��convex hull�� of its control points for 0 �£� t �£� 1.��'������@�Õ@�$��� €�€���œ€�‚ÿ���C������®@�A�,��� (€.�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Affine Invariance��9����Õ@�QB�%��� €)�€���ì€�‚ÿ�The Bézier curve is affinely invariant with respect to its control points. This means that any linear transformation (such as rotation or scaling) or translation of the control points defines a new curve that is just the transformation or translation of the original curve.��'������A�xB�$��� €�€���œ€�‚ÿ���G������QB�¿B�,��� (€6�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Variation Diminishing��¼���‹���xB�{C�1��� 0€�€���ì€�âÎ-«€�‰€�‚ÿ�The Bézier curve is variation diminishing. This means that it does not ��wiggle�� any more than its control polygon; it may wiggle less.��'������¿B�¢C�$��� €�€���œ€�‚ÿ���B������{C�äC�,��� (€,�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Linear Precision��ý���Ø���¢C�áD�%��� €±�€���ì€�‚ÿ�The Bézier curve has linear precision: If all the control points form a straight line, the curve also forms a line. This follows from the convex hull property; as the convex hull becomes a line, so does the curve.��ë���”���äC�ÌE�W��� |€+�€���œ€�‚‚È�ExecProgram("demo2.exe", 0)�€�‡"€��J�‰€�‚‚ÿ��������Click on this interactive application to study each of the characteristics of Bézier curves. Each tab selects a different characteristic.���1������áD�ýE�-��� *€ �€��œ€�†"€���‚ÿ����3��ê��ÌE�0H�I��� `€Õ�€���œ€�‚‚€ �€�‚‚€�€�€�€�âw…¨š€�‰€�‚‚‚ÿ����The de Casteljau Algorithm���Evaluation of the Bézier curve function at a given value t produces a point �f�(t). As t varies from 0 to 1, the point �f�(t) traces out the curve segment. One way to evaluate ��equation 3.5�� is by direct substitution, that is, by applying the value of t to the formula and computing the result.��This is probably the worst method of evaluating a point on the curve! Numerical instability, caused by raising small values to high powers, generates errors.��~��W��ýE�®I�'��� €¯�€���œ€�‚‚‚ÿ��There are several better methods available for evaluating the Bézier curve. One such method is the de Casteljau algorithm. This method not only provides a general, relatively fast, and robust algorithm, but it gives insight into the behavior of Bézier curves and leads to several important operations on the curves, such as the following:������_���0H�>J�1��� 2€¾�€��Pì:‚l€ �€�ƒ€ �€�‚ÿ�·���Computing derivatives. �The derivative of the curve gives the tangent vector at a point.��'������®I�eJ�$��� €�€���œ€�‚ÿ���<�� ��>J�¡K�2��� 2€�€��Pì:‚l€ �€�ƒ€ �€�‚ÿ�·���Subdividing the curve. �It is sometimes necessary to take a single Bézier curve and produce two separate curve segments that together are identical to the original. To accomplish this, it is necessary to find two sets of control points for the two new curves.��)��©��eJ�ÊM�€��� ΀U�€���œ€�‚‚‚È1�ExecProgram("demo1.exe Linear_Interpolation", 0)�€�‡"€��'�‰€�‚‚€�€�€�€�€�€�‚ÿ��The de Casteljau algorithm can be regarded as repeated linear interpolation.�������Use this interactive application, introduced in the topic on Preliminary Mathematics, to refresh the concept of linear interpolation with an independent parameter.��As described in the section on linear interpolation in Topic 2, "Preliminary Mathematics," it is possible to interpolate between two points �b�0� and �b�1� with the equation��'������¡K�ñM�$��� €�€���œ€�‚ÿ���?��� ���ÊM�0N�4��� 8€�€��0ì߀‚¡…€�†"€��K�ƒ‚ÿ���(3.10)��Q����ñM�O�8��� >€3�€���œ€�‚€�€�‚‚‚€ �€�‚‚‚‚ÿ��If t = 0.5, then �f�(t) is the midpoint of the line between the endpoints. Equation 3.10 is just a Bézier curve of degree n = 1.����De Casteljau's Algorithm for a Degree 2 Bézier Curve���This model may now be extended to handle a quadratic (degree 2) Bézier curve, as follows,���:������0N�»O�,��� (€�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Step One��{���M���O�B€�.��� ,€œ�€���œ€�‡"€��L�‚‚ÿ�� Consider the Bézier »O�B€�ýÉ�curve defined by three control points in the plane.���:������»O�|€�,��� (€�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Step Two��Ã���”���B€�?�/��� ,€+�€���œ€�‡"€��M�‚‚ÿ��For a specific value of t, interpolate between adjacent pairs of endpoints. In this example, t = 0.2. The resulting points are shown in yellow.���<������|€�{�,��� (€ �€��T˜î:‚l€ �€�ƒ‚ÿ�·��Step Three��Ù���ª���?�T‚�/��� ,€W�€���œ€�‡"€��N�‚‚ÿ��Using the same value of t (0.2), connect and interpolate between these two points. The resulting point is shown in green. This point is on the degree 2 Bézier curve.���;������{�‚�,��� (€�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Step Four��þ���Ï���T‚�ƒ�/��� ,€¡�€���œ€�‡"€��O�‚‚ÿ��By repeating this procedure using a series of values of t, a set of points on the Bézier curve is produced. The point produced by the application of de Casteljau's algorithm traces out the Bézier curve.���<������‚�Ƀ�,��� (€ �€��T˜î:‚l€ �€�ƒ‚ÿ�·��String Art��„��U��ƒ�M…�/��� ,€­�€���œ€�‡"€��P�‚‚ÿ��With finer granularity in the change in t, a picture similar to "string art," created by stretching string between nails on a board, is produced. Although the lines are straight, the boundary is a parabola, the Bézier curve. This process generalizes to Bézier curves of any degree; higher degree simply implies more levels of recursion.���]���1���Ƀ�ª…�,��� (€b�€��T˜î:‚l€ �€�ƒ‚ÿ�·��A Demonstration of de Casteljau's Algorithm��Î���j���M…�x†�d��� ˜€Ö�€���œÈ/�ExecProgram("demo1.exe deCasteljau_Bezier", 0)�€�‡"€��Q�‰€�‚‚ÿ�����Click on this interactive application to experiment with the behavior of de Casteljau's algorithm.���1������ª…�©†�-��� *€ �€��œ€�†"€���‚ÿ����#��ó���x†�̇�0��� .€ç�€���œ€�‚‚€ �€�‚‚‚‚ÿ����Labeling the Bézier Curve for de Casteljau's Algorithm���To formalize de Casteljau's algorithm, a labeling scheme is needed that includes the points produced by the process of recursive linear interpolation. The scheme works as follows:���R����©†�‰�O��� l€ �€��T˜î:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚†"€��R�‚€ �€�ƒ‚ÿ�·��Each level of recursion is denoted by a superscript.��·��The control points are the zeroth level and do not need to be superscripted.��·��Each successive level of recursion has one less point than the previous level.����·��For any point, it may be shown��@��� ���̇�^‰�5��� :€�€��T˜îÆ„l¡…€�†"€��S�ƒ‚ÿ���(3.11)��p�� ��‰�΋�P��� n€C�€���œ€�‚‚€ �€�‚‚€ �€�‚‚ë�Ê—Æù€�‡"€��T�‰€�‚ÿ����The Systolic Array: A Visualization of de Casteljau's Algorithm���One of the most important devices for dealing with curves is the �systolic array�. A systolic array is a triangular arrangement of vectors in which each row reflects the levels of recursion of the de Casteljau algorithm. The first row consists of the Bézier control points. Each successive row corresponds to the points produced by iterating with de Casteljau's algorithm.�������Click here for an interactive display of de Casteljau's algorithm, shown as a systolic array.��Ê���˜���^‰�˜Œ�2��� 2€3�€���œ€�‚†"€��U�‚‚‚‚ÿ�����Any point in the systolic array may be computed by linearly interpolating the two points in the preceding row with the parameter t; for example,���2������΋�ÊŒ�.��� ,€ �€��P�œÈ€�†"€��V�‚ÿ�������i���˜Œ�Y�&��� €Ò�€���œ€�‚‚‚ÿ��This process of linear interpolation is the fundamental operation in defining or evaluating a curve.���1������ÊŒ�Š�-��� *€ �€��œ€�†"€���‚ÿ����K����Y�Õ�J��� b€�€���œ€�‚‚€ �€�‚‚‚‚ë�c¨$&€�‡"€��W�‰€�‚ÿ����Subdivision of a Bézier Curve���One of the most important operations on a curve is that of subdividing it. The de Casteljau algorithm not only evaluates a point on the curve, it also subdivides a curve into two parts as a bonus. The control points of the two new curves appear along the sides of the systolic array. The new curves match the original in position, although they differ in parameterization.�������Click here to see the process of curve subdivision as the outcome of the de Casteljau algorithm.��H������Š�)À�-��� *€6�€���œ€�‚‚€ �€�‚Õ�)À�ýÉ�‚ÿ����Uses of Subdivision����g��0��Õ�Â�7��� <€a�€��T˜î:‚l€ �€�ƒ‚€ �€�ƒ‚ÿ�·��Design Refinement�Subdivision permits existing designs to be refined and modified. For example, additional curves may be incorporated into an object. This is accomplished by adding more control points for local control.��·��Clipping a Curve to a Boundary�One method of intersecting a Bézier curve with a line is to recursively subdivide the curve, testing for intersections of the curve's control polygons with the line. Curve segments not intersecting the line are discarded. This process is continued until a sufficiently fine intersection is attained.��ñ���Š���)À�Ã�g��� œ€�€���œ€�‚È-�ExecProgram("demo1.exe Subdivide_Bezier", 0)�€�‡"€��X�‰€�‚‚ÿ�������Click on this interactive application to see how a Bézier curve may be subdivided into two curves with de Casteljau's algorithm.���1������Â�²Ã�-��� *€ �€��œ€�†"€���‚ÿ����Z��ç��Ã� Æ�s��� ´€Ñ�€���œ€�‚‚€ �‚€�‚‚‚È/�ExecProgram("demo1.exe High_Degree_Bezier", 0)�€�‡"€��Y�‰€�‚ÿ����Higher Degree Bézier Curves���Bézier curves of any degree may be created. Degree 2 curves (quadratic curves) are the lowest degree useful. Many commercial applications and drawing packages use degree 3 Bézier curves (cubic curves) as a drawing primitive.�������Click on this interactive application to study Bézier curves of arbitrary degree. Use the left mouse button to add control points and increase the degree; use the right mouse button to delete points and lower the degree.��'������²Ã�3Æ�$��� €�€���œ€�‚ÿ���1������ Æ�dÆ�-��� *€ �€��œ€�†"€���‚ÿ������á���3Æ�uÇ�0��� .€Ã�€���œ€�‚‚€ �€�‚‚‚‚ÿ����The Derivative of the Bézier Curve���It is a straightforward exercise in algebra to differentiate the Bézier function and then to formulate the derivative in Bézier form. The derivative of a Bézier curve of degree n is���>��� ���dÆ�³Ç�3��� 6€�€��PœÐ‚¡…€�†"€��Z�ƒ‚ÿ���(3.12)��-����uÇ�àÈ�'��� € �€���œ€�‚‚‚ÿ��There is one less term in the derivative than in the original function. Also, the degree of the Bernstein polynomials is one less. The control points for the derivative curve are successive differences of the original curve's control points, having the form���2������³Ç�É�.��� ,€ �€��P�œÐ€�†"€��[�‚ÿ����i���C���àÈ�{É�&��� €†�€���œ€�‚‚‚ÿ��Consider the derivatives at the endpoints of the Bézier curve:���>��� ���É�¹É�3��� 6€�€��PœÐ‚¡…€�†"€��\�ƒ‚ÿ���(3.13)��†��[��{É�?Ë�+��� $€·�€���œ€�‚‚‚‚‚‚‚ÿ��where n is a constant.��The tangent endpoint property is derived from these two derivatives. It may be seen that the derivatives (the tangents) at the endpoints are n times the first and last legs of the control polygon.��The hodograph of the Bézier curve is easy to construct in Bézier form. It is a Bézier curve with control points given by���>��� ���¹É�}Ë�3��� 6€�€��PœÐ‚¡…€�†"€��]�ƒ‚ÿ���(3.14)����±���?Ë�•Ì�g��� œ€e�€���œ€�‚È-�ExecProgram("demo1.exe Hodograph_Bezier", 0)�€�‡"€��^�‰€�‚‚ÿ�������This interactive demonstration shows the correspondence between the Bézier curve and its hodograph. Try changing the control points and see how the hodograph responds.���1������}Ë�ÆÌ�-��� *€ �€��œ€�†"€���‚ÿ����ˆ��N��•Ì�NÏ�:��� B€Ÿ�€���œ€�‚‚€ �€�‚‚‚‚‡"€��_�‚ÿ����Continuity of the Bézier Curve���Earlier discussions on continuity focused on the differences between C1 and G1 continuity. The following figure shows a curve that is C1 but not G1; a particle tracing out the path of the curve would undergo a sudden reversal of direction at the kink, so the curve is not G1. However, its hodograph is smooth, which makes the curve C1. ���Notice how the derivative goes to zero at the kink of the curve. This permits a smooth hodograph, without a smooth curve. Clearly, the superior measure of smoothness provided by geometric continuity is desirable.��Õ��£��ÆÌ�/�2��� 2€I�€���œ€�‚‚‚‡"€��`�‚‚ÿ��Consider now the composite curve shown below. There are two Bézier curves attached at the point A. The composite curve is not NÏ�/�ýÉ�C1 at A since the derivative jumps in magnitude. The direction of the derivative does not change, however, and its path is smooth; it is G1.���The tangent vector (derivative) of the composite curve undergoes a discontinuous step in magnitude at the point where the two Bézier curves join.���1������NÏ�`�-��� *€ �€��œ€�†"€���‚ÿ����Û���«���/�;�0��� .€W�€���œ€�‚‚€ �€�‚‚‚‚ÿ����Degree Elevation of the Bézier Curve���Any polynomial of degree n can be thought of as having higher degree terms than n if the coefficients are zero. For instance,���2������`�m�.��� ,€ �€��P�¤È€�†"€��a�‚ÿ������Ü���;�p�'��� €¹�€���œ€�‚‚‚ÿ��This may seem pointless to do in standard polynomial form, but if the polynomial is in Bézier form, the coefficients of the higher degree polynomial are generally not zero. That means, for example, that the parabola���>��� ���m�®�3��� 6€�€��PœÐ‚¡…€�†"€��b�ƒ‚ÿ���(3.15)��<������p�ê�&��� €,�€���œ€�‚‚‚ÿ��has a cubic form:���>��� ���®�(�3��� 6€�€��PœÐ‚¡…€�†"€��c�ƒ‚ÿ���(3.16)�� ��Å��ê�2�E��� X€�€���œ€�‚‚‚‡"€��d�‚‚€�€�€�€�€�€�‚ÿ��This is identical in shape and in parameterization to the quadratic form.���This figure shows the quadratic and cubic forms for the same Bézier curve. The degree of the curve has been elevated without changing its shape or parameterization.��This means that if the cubic Bézier curve in the figure above is written as a standard polynomial, it would have one zero coefficient (�a�3�), but in Bézier form, none of the control points (�b�i�) are zero.��§���{���(�Ù�,��� (€ö�€���œ€�‚€ �€�‚‚ÿ��The algorithm to find the Bézier curve of the next higher degree is called �degree elevation�. It proceeds as follows.���û���¸���2�Ô�C��� T€q�€��˜œ‚H€�€�€�€�€�€�€�€�€�€�‚ÿ�Given a Bézier curve of degree n with control points �b�0�, �b�1�, ..., �b�n�, it may be proved that the control points for the Bézier curve of degree n + 1 can be found as follows:��7������Ù� �2��� 4€ �€��œ‚H€ �†"€��e�€�‚ÿ�����'������Ô�2�$��� €�€���œ€�‚ÿ���=��� ��� �o�2��� 4€�€��œ‚¡…€�†"€��f�ƒ‚ÿ���(3.17)��V��ï���2�Å �g��� œ€á�€���œ€�‚È+�ExecProgram("demo1.exe Elevate_Bezier", 0)�€�‡"€��g�‰€�‚‚ÿ�������Click on this interactive application to see how degree elevation preserves the curve shape while adding control points.��Note particularly how the control polygon begins to take the shape of the curve as the degree is increased.���1������o�ö �-��� *€ �€��œ€�†"€���‚ÿ����§��w��Å � �0��� .€ï�€���œ€�‚‚€ �€�‚‚‚‚ÿ����What Has Been Accomplished in This Topic���The Bézier curve has been covered in depth, with discussions of the properties of the curve. De Casteljau's algorithm generates the curve using iterated linear interpolation, providing a fast and stable way of creating Bézier curves. Subdivision, degree elevation, and differentiation of the Bézier curve completed the topic.���1������ö �Î �-��� *€ �€��œ€�†"€���‚ÿ����)������ �÷ �%��� €�€���œ€�‚‚ÿ����Ç������Î �¾ �H��� `€þ�€��0�¤¨ã¤“h€�‰€�‚ã 8ûU€�‰€�‚㋼ŸÞ€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: Interpolation�����Go to the previous topic: Preliminary Mathematics����B������÷ �� �1���z��ÿÿÿÿÿÿÿÿ$���� �; �„�Curve Subdivision;������¾ �; �'��� €(�€���˜˜B˜€ �‚ÿ�Curve Subdivision��'������� �b �$��� €�€���œ€�‚ÿ���1������; �“ �-��� *€ �€��œ€�†"€��h�‚ÿ����¡���u���b �4�,��� (€ê�€���œ€�‚€�‚€�‚ÿ���This is a cubic Bézier curve after three iterations of the de Casteljau algorithm, with the parameter t = 0.5.����P������“ �„�5��� :€6�€��pì¨:‚lë�d¨$&€�‰€�‚ÿ��After the subdivision����B������4�Æ�1���õ��ÿÿÿÿÿÿÿÿ%���Æ��¹@�Curve Subdivision;������„��'��� €(�€���˜˜B˜€ �‚ÿ�Curve Subdivision��'������Æ�(�$��� €�€���œ€�‚ÿ���1�������Y�-��� *€ �€��œ€�†"€��i�‚ÿ����ï���Â���(�T@�-��� (€…�€���œ€�‚€�‚€�‚ÿ���By using the left and right legs of the systolic array as control points, two separate Bézier curves are obtained that Y�T@�„�together replicate the original. The curve is subdivided at t = 0.5.����e������Y�¹@�G��� ^€<�€��pì¨:‚lÈ�CloseWindow("SysArray")�€�‰€�‚ÿ��Back to the Bézier curve����?������T@�ø@�1�����ÿÿÿÿÿÿÿÿ&���ø@�4A�B�Systolic Array<������¹@�4A�*��� $€$�€���˜˜B˜€ �€�‚ÿ�Systolic Array���'������ø@�[A�$��� €�€���œ€�‚ÿ���1������4A�ŒA�-��� *€ �€��œ€�†"€��j�‚ÿ����'������[A�³A�$��� €�€���œ€�‚ÿ���S������ŒA�B�5��� :€<�€��pì¨:‚lë�Ë—Æù€�‰€�‚ÿ��Go to the next iteration����?������³A�EB�1�����ÿÿÿÿÿÿÿÿ'���EB�B�SC�Systolic Array<������B�B�*��� $€$�€���˜˜B˜€ �€�‚ÿ�Systolic Array���'������EB�¨B�$��� €�€���œ€�‚ÿ���1������B�ÙB�-��� *€ �€��œ€�†"€��k�‚ÿ����'������¨B��C�$��� €�€���œ€�‚ÿ���S������ÙB�SC�5��� :€<�€��pì¨:‚lë�Ì—Æù€�‰€�‚ÿ��Go to the next iteration����?�������C�’C�1�����ÿÿÿÿÿÿÿÿ(���’C�ÊC�œD�Systolic Array8������SC�ÊC�'��� €"�€���˜˜B˜€ �‚ÿ�Systolic Array��'������’C�ñC�$��� €�€���œ€�‚ÿ���1������ÊC�"D�-��� *€ �€��œ€�†"€��l�‚ÿ����'������ñC�ID�$��� €�€���œ€�‚ÿ���S������"D�œD�5��� :€<�€��pì¨:‚lë�Í—Æù€�‰€�‚ÿ��Go to the next iteration����?������ID�ÛD�1�����ÿÿÿÿÿÿÿÿ)���ÛD�E�úE�Systolic Array8������œD�E�'��� €"�€���˜˜B˜€ �‚ÿ�Systolic Array��'������ÛD�:E�$��� €�€���œ€�‚ÿ���1������E�kE�-��� *€ �€��œ€�†"€��m�‚ÿ����'������:E�’E�$��� €�€���œ€�‚ÿ���h���!���kE�úE�G��� ^€B�€��pì¨:‚lÈ�CloseWindow("SysArray")�€�‰€�‚ÿ��Go back to the Bézier curve����M������’E�GF�1���Ë���ÿÿÿÿÿÿÿÿ*���GF�ÿÿÿÿÅF�Equation 5: The Bézier curveL���"���úE�“F�*��� $€D�€���˜˜B˜€ �€�‚ÿ�Equation 3.5: The Bézier Curve���2������GF�ÅF�.��� ,€ �€��P�¦Æ€�†"€��n�‚ÿ����:��� ���“F�ÿF�1���G��ÿÿÿÿÿÿÿÿ+���ÿF�ÿÿÿÿ H�B-Splines7��� ���ÅF�6G�*��� $€�€���˜˜B˜€ �€�‚ÿ�B-Splines���Ö���¯���ÿF� H�'��� €_�€���¦€�‚ÿ�A B-spline is a highly controllable spline curve, a favorite�of industrial designers. It is a piecewise polynomial, which�may be described as a collection of Bézier curves.��@������6G�LH�1�����ÿÿÿÿÿÿÿÿ,���LH�ÿÿÿÿI�Surface Patches=������ H�‰H�*��� $€&�€���˜˜B˜€ �€�‚ÿ�Surface Patches���Š���e���LH�I�%��� €Ê�€���¦€�‚ÿ�Surface patches are three-dimensional surface sections�that may be combined to form solid objects.��G������‰H�ZI�1���S��ÿÿÿÿÿÿÿÿ-���ZI�ÿÿÿÿfK�Parabola ReformulationB������I�œI�*��� $€0�€���˜˜B˜€ �€�‚ÿ�Parabola Reformation���Q���-���ZI�íI�$��� €Z�€���œ€�‚ÿ�This parabola is defined parametrically by��4������œI�!J�0��� 0€ �€��V�ŒŒ¦Æ€�†"€��o�‚ÿ����,������íI�MJ�&��� € �€���ŒŒ¦€�‚ÿ�and��8������!J�…J�3��� 6€ �€��V�ŒŒ¦Æ€�†"€��p�€�‚ÿ�����O���)���MJ�ÔJ�&��� €R�€���ŒŒ¦€�‚ÿ�In vector form, it may be expressed as��4������…J�K�0��� 0€ �€��V�ŒŒ¦Æ€�†"€��q�‚ÿ����^���:���ÔJ�fK�$��� €t�€���œ€�‚ÿ�The significance of the coefficients is explained next.��<��� ���K�¢K�1���`��ÿÿÿÿÿÿÿÿ.���¢K�ÿÿÿÿÆL�Convex Hull9������fK�ÛK�*��� $€�€���˜˜B˜€ �€�‚ÿ�Convex Hull���º���’���¢K�•L�(��� €%�€���˜¦€�‚ÿ�The convex hull of a control polygon is the minimal�convex enclosure of the control polygon.�In this picture, it is shown as a magenta outline.��1������ÛK�ÆL�-��� *€ �€���¦€�†"€��r�‚ÿ����F������•L� M�1���e��ÿÿÿÿÿÿÿÿ/��� M�ÿÿÿÿ+N�Variation DiminishingP���&���ÆL�\M�*��� $€L�€���˜˜B˜€ �€�‚ÿ�"Wiggles" in Variation Diminishing���Ï���§��� M�+N�(��� €O�€���¦€�‚ÿ�"Wiggle" means the way in which a curve or surface�changes direction. This is more precisely expressed�as a change in sign of the curvature of the curve or�surface.��J������\M�uN�1��� ��ÿÿÿÿÿÿÿÿ0���uN�ÿÿÿÿ €�The Binomial CoefficientsG������+N�¼N�*��� $€:�€���˜˜B˜€ �€�‚ÿ�The Binomial Coefficients���}���X���uN�9O�%��� €°�€���œ€�‚‚ÿ�The binomial coefficients, commonly derived from Pascal's triangle, may be computed:���=��� ���¼N�vO�3��� 6€�€��PœÈ‚¡…€�†"€��s�ƒ‚ÿ���(3.7)��U������9O� €�6��� <€@�€���œ€�‚€ �€�‚‚†"€��t�‚ÿ��for i, an integer �³� 0.����������������������������������������������������������vO� €�+N�H������vO�T€�1���Á��ÿÿÿÿÿÿÿÿ1���T€�š€�*†�The Bernstein FunctionsF������ €�š€�+��� &€6�€��^�˜˜B˜Œ€ �€�‚ÿ�The Bernstein Functions���´������T€�N�'��� €�€���¤€�‚‚‚ÿ��The Bernstein functions were originally devised by Bernstein to prove the famous Weierstrass theorem in 1912. They are formally given by���=��� ���š€�‹�3��� 6€�€��PœÈ‚¡…€�†"€��u�ƒ‚ÿ���(3.8)��%��º��N�°ƒ�k��� ¤€w�€���¤€�‚‚‚È,�ExecProgram("demo1.exe Bernstein_Basis", 0)�€�‡"€��v�‰€�‚‚‚ÿ��They have many useful properties for curve generation.�������Click on this interactive application to gain a deeper understanding of the Bernstein basis functions.��When the application is running, use the "Degree" button to change the degree of the Bernstein polynomial.��An important characteristic of the Bernstein functions is the partition of unity. This simply means that the sum of the functions is always one, for all values of t:��'������‹�׃�$��� €�€���¤€�‚ÿ���=��� ���°ƒ�„�3��� 6€�€��PœÈ‚¡…€�†"€��w�ƒ‚ÿ���(3.9)��ú���Ê���׃�…�0��� .€•�€���œ€�‚‚€ �‚€�‚‚‚ÿ����Exercise���Can you show that the Bernstein functions form a basis? That is, can you write any polynomial in Bernstein form? Pick a set of Bernstein polynomials, for example, with n = 3, then show���2������„�@…�.��� ,€ �€��P�œÐ€�†"€��x�‚ÿ����U���"���…�•…�3��� 6€D�€���œ€�‚âr~È€�‰€�‚‚‚ÿ��for any value of t. ��Hint������1������@…�Æ…�-��� *€ �€��œ€�†"€���‚ÿ����d���3���•…�*†�1��� 2€f�€���œ€�‚ãk©:"€�‰€�‚ÿ����Return to the Introduction to Bézier Curves����5������Æ…�_†�1���Û���ÿÿÿÿÿÿÿÿ2���_†�ÿÿÿÿ‡�Hint.������*†�†�'��� €�€���˜˜B˜€ �‚ÿ�Hint��x���S���_†�‡�%��� €¦�€���œ€�‚ÿ�Start with the simple quadratic form, and�extend the solution to a general form.��>��� ���†�C‡�1���<@��äƒ�wƒ �3���C‡�ƒ‡�ă�Interpolation@������‡�ƒ‡�'��� €2�€���˜˜B¤€ �‚ÿ�Topic 4: Interpolation��F��� ���C‡�ɇ�&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn����Á���ƒ‡�ψ�E��� X€ƒ�€��2˜¤ª‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��why interpolation is important,��·��how Lagrange interpolation works,��·��the use of Aitken's algorithm for computing interpolating curves,��·��the Hermite interpolant and its advantages.��'������ɇ�öˆ�$��� €�€���¤€�‚ÿ���2������ψ�(‰�.��� ,€ �€��0¤¨€�†"€���‚ÿ����«��d��öˆ�Ó‹�G��� \€É�€���¤€�‚‚€ �‚€�‚â("Œ€�€�‰âú…9]€�‰€�‚ÿ����Background to Interpolation���To ��interpolate� �in mathematics means to estimate values between given known values. The Bézier curve, for instance, interpolates values between its endpoints. The other control points are not usually interpolated by the curve. There is, however, a polynomial form that interpolates all of the control points. It is the ��Lagrange ��form. The Lagrange form is a good example of the more general topic of interpolation. It has connections to both iterated linear interpolation (for example, de Casteljau's algorithm) and nonuniform B-splines, which are introduced in Topic 6.��§��6��(‰�z�q��� °€o�€���¤€�‚È3�ExecProgram("demo3.exe Lagrange_Interpolation", 0)�€�‡"€��y�‰€�‚‚‚‚ÿ�������Click on the graphic for a demonstration of a cubic Lagrange interpolation curve.��Use the left mouse button to drag the data points, and observe the behavior of the Lagrange curve.��Based upon the discussion of the Bézier curve in Topic 3, the expected form of the Lagrange interpolation function is���=��� ���Ó‹�·�3��� 6€�€��P¤È‚¡…€�†"€��z�ƒ‚ÿ���(4.1)��ø��¬��z�»À�L��� f€Y�€���¤€�‚€�€�âìGNZ€�‰€�‚‚€�€�€�€�€�€�‚ÿ��for some points �c� (to be interpolated) and some ��basis functions�� L(t). From the study of Bézier curves and Bernstein basis functions, the Lagrange basis functions should rightly be expected to be the key to understanding the interpolation behavior.��In the Bézier case, interpolation of the endpoints occurred because at the parameter values corresponding to the endpoints (t varies from 0 to 1) all the basis functions are zero except the first and last, which are one. Thus �f�(0) and �f�(1) are exac·�»À�‡�tly the control points. Applying this to the Lagrange case, basis functions are sought so that the ith basis function at some parameter t�i� is one, and all others are zero.��§���{���·�bÁ�,��� (€ö�€���¤€�‚€�€�‚‚ÿ��In order to achieve this, a particular value of the parameter t must be associated with each point to interpolate �c�:���2������»À�”Á�.��� ,€ �€��P�¤È€�†"€��{�‚ÿ����‚���V���bÁ�Â�,��� (€¬�€���¤€�‚€ �€�‚‚ÿ��These values of the parameter t are called �knots� and may be selected as long as���2������”Á�HÂ�.��� ,€ �€��P�¤È€�†"€��|�‚ÿ����„���^���Â�ÌÂ�&��� €¼�€���¤€�‚‚‚ÿ��Now consider the basis function that will give the interpolation property for all points:���=��� ���HÂ� Ã�3��� 6€�€��P¤È‚¡…€�†"€��}�ƒ‚ÿ���(4.2)��Î���›���ÌÂ�×Ã�3��� 4€7�€���¤€�‚€�€�€�€�‚‚ÿ��Notice that the numerator is the product of terms (t�k� - t), except where k = i. For any t = t�k� (k is not equal to i), the numerator is zero. Thus,���2������ Ã� Ä�.��� ,€ �€��P�¤È€�†"€��~�‚ÿ����¼������×Ã�ÅÄ�-��� (€�€���¤€�‚€�€�‚‚ÿ��This is part of what is sought. The other part follows by noting that if t = t�i�, then terms in both numerator and denominator cancel and���2������ Ä�÷Ä�.��� ,€ �€��P�¤È€�†"€���‚ÿ����Z���4���ÅÄ�QÅ�&��� €h�€���¤€�‚‚‚ÿ��This is exactly what is needed; this means that���2������÷Ä�ƒÅ�.��� ,€ �€��P�¤È€�†"€��€�‚ÿ����¡���{���QÅ�$Æ�&��� €ö�€���¤€�‚‚‚ÿ��This can be seen graphically in the following interactive demonstration, which shows a cubic Lagrange basis with knots���2������ƒÅ�VÆ�.��� ,€ �€��P�¤È€�†"€���‚ÿ����ù��‡��$Æ�OÈ�r��� ²€�€���¤€�‚È+�ExecProgram("demo3.exe Lagrange_Basis", 0)�€�‡"€��‚�‰€�‚‚‚‚‚€ �€�‚‚ÿ�������Click on the graphic for a demonstration of the cubic Lagrange basis functions.��Use the left mouse button to drag the knot points, and observe the behavior of the cubic Lagrange basis functions.��Notice how the basis functions are one at exactly one knot and zero at the others. This is the condition for interpolation.����Comparison of the Bernstein and Lagrange Basis Functions����H������VÆ�—È�,��� (€8�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Endpoint Interpolation��Ó���•���OÈ�jÉ�>��� J€+�€���˜î€�€�€�€�€�€�€�€�€�‚ÿ�Where the Bézier curve interpolates the first and last points �b�0� and �b�n�, the Lagrange basis causes interpolation of every control point �c�.��'������—È�‘É�$��� €�€���¤€�‚ÿ���D������jÉ�ÕÉ�,��� (€0�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Tangent Conditions��Ï���ª���‘É�¤Ê�%��� €U�€���ì€�‚ÿ�The Bézier curve is tangent to the first and last segments of the control polygon, at the first and last control points. The Lagrange form does not have this property.��'������ÕÉ�ËÊ�$��� €�€���¤€�‚ÿ���C������¤Ê�Ë�,��� (€.�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Affine Invariance��#��þ���ËÊ�1Ì�%��� €ý�€���ì€�‚ÿ�The Lagrange interpolant is affinely invariant with respect to its control points. This means that any linear transformation or translation of the control points defines a new curve that is just the transformation or translation of the original curve.��'������Ë�XÌ�$��� €�€���¤€�‚ÿ���=������1Ì�•Ì�,��� (€"�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Convex Hull��\����XÌ�ñÍ�I��� `€'�€���ì€�âÒýÔ#€�‰€�€ �€�€ �€�â†3kl€�‰€�‚ÿ�The Bézier curve is contained in the ��convex hull�� of its control points for 0 �£� t �£� 1. Since the Lagrange basis does not possess the ��partition of unity�� property, and because the basis functions may take negative values, the convex hull condition does not apply.��'������•Ì�Î�$��� €�€���¤€�‚ÿ���G������ñÍ�_Î�,��� (€6�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Variation Diminishing��s���C���Î�ÒÎ�0��� 0€†�€���ì€�âà¶æX€�‰€�‚ÿ�The Lagrange interpolant curve is not ��variation diminishing.����'������_Î�ùÎ�$��� €�€���¤€�‚ÿ���B������ÒÎ�;Ï�,��� (€,�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Linear Precision��º���•���ùÎ� ��%��� €+�€���ì€�‚ÿ�Like the Bézier curve, the Lagrange interpolant has linear precision: If all the control points form a straight line, the curve also forms a line.�������������;Ï� ��‡�„��\��;Ï��(��� €¹�€���¤€�‚‚‚‚ÿ��Despite the limitations of the Lagrange basis functions, they may be appropriate for some applications.��Before leaving the topic of Lagrange basis functions, there is something special to note: the basis functions depend on a knot sequence for definition. This was not true of the Bernstein polynomials; they are independent of any relation between control points and parameter space. In the following interactive demonstration, selection of the knots can change the shape of the basis functions and the manner in which the interpolating curve responds. This will be seen again with B-spline curves.��G��à�� ��×�g��� œ€Ã�€���¤€�‚È*�ExecProgram("demo3.exe Lagrange_Demo", 0)�€�‡"€��ƒ�‰€�‚‚‚ÿ�������Click on the graphic for a demonstration of the cubic Lagrange interpolating curve with a nonuniform knot sequence.��Use the left mouse button to drag the data points or the knot points, and observe the behavior of the interpolating curve.��The shape of the curve can sometimes be quite difficult and unintuitive to control in the Lagrange form, especially compared to Bézier. Lagrange is not often used in shape design, but it provides a working method of interpolation.��=��������-��� *€ �€���¤€�‚‚€ �€�‚‚ÿ����Exercise�������[���×�£�4��� 8€¶�€��Pì:‚l€�ƒâQüé€�‰€�‚ÿ�1.�Can you give an example where interpolation is more important than design? ��Answer����'�������Ê�$��� €�€���¤€�‚ÿ���2������£�ü�.��� ,€ �€��0¤¨€�†"€���‚ÿ����‘��Z��Ê��7��� <€µ�€���¤€�‚‚€ �€�‚‚‚‚€ �€�‚ÿ����Evaluation of the Lagrange Curve���As in the case of Bézier curves, direct evaluation of a Lagrange interpolating curve is a poor approach; in fact, it's even worse than Bézier. Numerical instability is a major problem with direct evaluation of Lagrange; examine the basis functions L(t) and look for places where division by zero may occur. Use of very small values is also a challenge.��Iterated linear interpolation again becomes useful. The analog of the de Casteljau algorithm in the case of Lagrange curves is the �Aitken algorithm�. Starting with a quadratic case, let the knot sequence be��'������ü�´�$��� €�€���¤€�‚ÿ���2�������æ�.��� ,€ �€��P�¤È€�†"€��„�‚ÿ����0��¬��´� �„��� Ö€c�€���¤€�‚‚‚‡"€��…�‚‚€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��†�‚‚†"€��‡�‚†"€��ˆ�‚‚‡"€��‰�‚‚‚ÿ��The following figure shows the interpolation points and the knot sequence:�����Find the point on the curve at t = 0.5. By linear interpolation from �c�0� to �c�1�, a point is found on the extension of �c�0�-�c�1�, resulting in the following figure:�������������The pattern of iterated interpolation should be evident. Something changed from the first iteration to the second; examine the following interactive demonstration.��±��D��æ�Ç �m��� ¨€‹�€���¤€�‚È/�ExecProgram("demo3.exe Quadratic_Lagrange", 0)�€�‡"€��Š�‰€�‚‚‚‚ÿ�������Click on the graphic for a demonstration of the quadratic Lagrange interpolating curves.��Use the left mouse button to drag the data points, and observe the behavior of the interpolating curve.��In this interactive demonstration, a uniform knot sequence is used. It is easy to see that the first iteration computes���=��� ��� � �3��� 6€�€��P¤È‚¡…€�†"€��‹�ƒ‚ÿ���(4.3)��B������Ç �F �&��� €8�€���¤€�‚‚‚ÿ��It is less obvious that���=��� ��� �ƒ �3��� 6€�€��P¤È‚¡…€�†"€��Œ�ƒ‚ÿ���(4.4)��š���t���F ��&��� €è�€���¤€�‚‚‚ÿ��The second iteration uses a different knot-dependent weight than the first. In general, the Aitken algorithm is���=��� ���ƒ �Z�3��� 6€�€��P¤È‚¡…€�†"€���ƒ‚ÿ���(4.5)��½��T���#A�i���  €«�€���¤€�‚‚‚‚‚È*�ExecProgram("demo3.exe Lagrange_Demo", 0)�€�‡"€��ƒ�‰€�‚ÿ��The derivation of the points in Aitken's algorithm can be arranged in a systolic array. The difference between the de Casteljau array and Aitken's is that the weights change at each level in Aitken's. The weights are constant in de Casteljau's algorithm.��Now try the cubic Lagrange demonstration again. Change the Z�#A�‡�knots and observe the effects.�������Click on the graphic for a demonstration of the cubic Lagrange interpolating curves with a nonuniform knot sequence.��Use the left mouse button to drag the data points or the knot points, and observe the behavior of the interpolating curve.��'������Z�JA�$��� €�€���¤€�‚ÿ���2������#A�|A�.��� ,€ �€��0¤¨€�†"€���‚ÿ����~��@��JA�úB�>��� J€�€���¤€�‚‚€ �€�‚‚âýˆ ´€�‰€�‚‚‚‚ÿ����Hermite Interpolation���There is a very special interpolation form that is defined only for cubic curves. ��Hermite ��curves interpolate to two points and two given tangent vectors. This can be useful for design and interpolation problems where slopes are given or can be estimated.��The Hermite curve is given by���=��� ���|A�7C�3��� 6€�€��P¤È‚¡…€�†"€��Ž�ƒ‚ÿ���(4.6)�� ���n���úB�×C�2��� 4€Ü�€���¤€�‚€�€�€�€�‚‚ÿ��where �p� denotes the two endpoints and �m� the tangent vectors. The Hermite basis functions are given by���=��� ���7C�D�3��� 6€�€��P¤È‚¡…€�†"€���ƒ‚ÿ���(4.7)��z����×C�ŽE�t��� ¶€�€���¤€�‚âêû¶#€�‰€�‚‚È*�ExecProgram("demo3.exe Hermite_Basis", 0)�€�‡"€���‰€�‚‚‚‚ÿ��Here, the Hermite basis functions are given in terms of the ��Bernstein basis functions.���������Click on the graphic for a demonstration of the four cubic Hermite basis functions.��As with the other forms, at t = 0 all functions are zero, except the first:���2������D�ÀE�.��� ,€ �€��P�¤È€�†"€��‘�‚ÿ����j���D���ŽE�*F�&��� €ˆ�€���¤€�‚‚‚ÿ��This ensures the endpoint interpolation requirement. Similarly,���2������ÀE�\F�.��� ,€ �€��P�¤È€�†"€��’�‚ÿ����Ç��� ���*F�#G�'��� €A�€���¤€�‚‚‚ÿ��which ensures that the curve interpolates the second endpoint. A unique feature of Hermite is that the derivatives are all zero at t = 0, except the first:���2������\F�UG�.��� ,€ �€��P�¤È€�†"€��“�‚ÿ���� ��Ä���#G�^H�E��� X€‰�€���¤€�‚€�€�€�€�€�€�€�€�€�€�‚‚ÿ��This means that at the first point �h�(0), the Hermite curve's derivative is defined entirely by the first tangent vector �m�0�. Similarly, at �h�(1) the derivative is found from �m�1�, since���2������UG�H�.��� ,€ �€��P�¤È€�†"€��”�‚ÿ����L��Ý��^H�ÜJ�o��� ¬€½�€���¤€�‚‚‚È2�ExecProgram("demo3.exe Hermite_Interpolation", 0)�€�‡"€��•�‰€�‚ÿ��When t = 1, the other three basis functions have zero derivatives. The following interactive demonstration animates the Hermite form and allows you to swap with the Bézier form. Notice the similarities and the differences.�������Click on the graphic for a demonstration of the cubic Hermite curve.��Use the left mouse button to drag the control points and tangent vectors, or switch between Bézier and Hermite forms. View the basis functions by clicking the "Basis" button.��A����H�L�:��� B€�€���¤€�‚‚‚‡"€��–�‚‚‚€ �€�‚ÿ��The Hermite form is convenient for interpolating data with slope information. The cubic pieces can be attached with shared endpoints and tangent vectors to create a smooth curve through a data set. This is illustrated in the following figure:�������Exercises���Ô��� ���ÜJ�ñL�4��� 6€A�€��r˜ìª:‚l€�ƒ€�€�€�‚ÿ�2.�What is the relation between the first tangent vector (at �p�0�) in a Hermite curve and the first leg of the corresponding Bézier curve's control polygon?��>��� ���L�/M�1��� 2€�€��2˜ìª‚lâL/ €�‰€�‚ÿ��Answer.����¹���‚���ñL�èM�7��� <€�€��r˜ìª:‚l€�ƒ€�€�€�€�‚ÿ�3.�How would the �m� tangent vectors be selected if only a set of �p� endpoints is given? A smooth interpolant must be created.��>��� ���/M�&N�1��� 2€�€��2˜ìª‚lⵇԎ€�‰€�‚ÿ��Answer.����9������èM�_N�*��� $€�€��r˜ìª:‚l€�ƒ‚ÿ�4.�Show that��7������&N�–N�3��� 6€ �€��r˜¤ª‘€‚l€�†"€��—�‚ÿ����>��� ���_N�ÔN�1��� 2€�€��2˜ìª‚lâàœ€�‰€�‚ÿ��Answer.����^���4���–N�2O�*��� $€h�€��r˜ìª:‚l€�ƒ‚ÿ�5.�What properties does the Hermite form possess?��>��� ���ÔN�pO�1��� 2€�€��2˜ìª‚lâ‡8e€€�‰€�‚ÿ��Answer.����'������2O�—O�$��� €�€���¤€�‚ÿ���2������pO�ÉO�.��� ,€ �€��0¤¨€�†"€���‚ÿ����r��<��—O�G�6��� :€y�€���¤€�‚‚€ �€�‚‚€�€�‚‚ÿ�ÉO�G�‡����Other Interpolation Forms���There are other forms of interpolating curves. An important form that is beyond the scope of this tutorial is the interpolatory spline. It allows piecewise interpolation of data sets like Hermite, but with the constraint that the joins are C�n-1� continuous, where n is the degree.���2������ÉO�y�.��� ,€ �€��0¤¨€�†"€���‚ÿ������Ý���G�‰‚�3��� 4€»�€���¤€�‚‚€ �€�‚€�‚‚‚ÿ����What Has Been Accomplished in This Topic����Two methods for producing interpolating curves have been discussed, providing a solution to a common problem: that of passing a smooth curve through a set of data points.���2������y�»‚�.��� ,€ �€��0¤¨€�†"€���‚ÿ����)������‰‚�ä‚�%��� €�€���¤€�‚‚ÿ����¹���o���»‚�ƒ�J��� d€Þ�€��0¤¨‚l㤓h€�‰€�‚ãS‹FZ€�‰€�‚ã=Ë4h€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: Blossoms�����Go to the last topic: The Bézier Curve����'������ä‚�ă�$��� €�€���¤€�‚ÿ���<��� ���ƒ��„�1���/��ÿÿÿÿÿÿÿÿ4����„�ÿÿÿÿó„�Interpolate9������ă�9„�*��� $€�€���˜˜B¤€ �€�‚ÿ�Interpolate���º���ˆ����„�ó„�2��� 2€�€���¤€�€ �€�€ �€�‚ÿ�From the Latin �inter� (between) and �polare� (to polish): to enlarge�(a book or manuscript) by insertion of words or subject matter.��9������9„�,…�1���è���ÿÿÿÿÿÿÿÿ5���,…�ÿÿÿÿÛ…�Lagrange2��� ���ó„�^…�'��� €�€���˜˜B¦€ �‚ÿ�Lagrange��}���W���,…�Û…�&��� €®�€���B˜€�‚ÿ�Count Louis de Lagrange, 1736-1813, was a noted French�astronomer and mathematician.��6������^…�†�1���|��ÿÿÿÿÿÿÿÿ6���†�ÿÿÿÿW‡�Bases;������Û…�L†�(��� €&�€���B˜€ �€�‚ÿ�Basis functions��� ��à���†�W‡�+��� $€Á�€���¤€�€�‚ÿ�Recall that given a space S of functions (a collection of functions�such as polynomials, trigonometric, etc.), a set B is a basis if all�functions of S are combinations of functions from B, and B�is as small as possible.���C������L†�š‡�1���Š��ÿÿÿÿÿÿÿÿ7���š‡�ÿÿÿÿáˆ�Partition of Unity@������W‡�Ú‡�*��� $€,�€���˜˜B˜€ �€�‚ÿ�Partition of Unity���L���'���š‡�&ˆ�%��� €N�€���¤€�‚‚ÿ�Recall for the Bernstein functions:���4������Ú‡�Zˆ�/��� .€ �€��P�œÈ€�†"€��˜�‚‚ÿ�����-������&ˆ�‡ˆ�&��� €�€��P�œˆ€�‚‚ÿ�and���2������Zˆ�¹ˆ�.��� ,€ �€��P�œÐ€�†"€��™�‚ÿ����(������‡ˆ�áˆ�%��� €�€��P�œÈ€�‚ÿ���<��� ���¹ˆ�‰�1���‘��ÿÿÿÿÿÿÿÿ8���‰�ÿÿÿÿrŠ�Convex Hull9������áˆ�V‰�*��� $€�€���˜˜B˜€ �€�‚ÿ�Convex Hull���ç���¿���‰�=Š�(��� €�€���˜¦€�‚ÿ�Recall that the convex hull of a control polygon is the minimal�enclosure of the control polygon such that if two points are in�the hull, then the line segment between is also in the hull.��5������V‰�rŠ�0��� 0€ �€���¦€�†"€��r�€�‚ÿ�����F������=Š�¸Š�1���D��ÿÿÿÿÿÿÿÿ9���¸Š�ÿÿÿÿ¶‹�Variation DiminishingC������rŠ�ûŠ�*��� $€2�€���˜˜B˜€ �€�‚ÿ�Variation Diminishing���»���‘���¸Š�¶‹�*��� "€#�€���¦€�€�‚ÿ�The Lagrange interpolant may well wiggle more than�its control polygon. "Wiggle" means the way in which�a curve or surface changes direction.���I������ûŠ�ÿ‹�1���’��ÿÿÿÿÿÿÿÿ:���ÿ‹�ÿÿÿÿH�Interpolation vs. DesignF������¶‹�EŒ�*��� $€8�€���˜˜B˜€ �€�‚ÿ�Interpolation vs. Design�����×���ÿ‹�H�,��� &€¯�€���¦€�€�‚ÿ�One example is slalom skiing, where the prime�objective is to "interpolate" the flags down the�slope. Another example is croquet, where the�ball must go through the hoops, regardless of�any other moves it makes.���@������EŒ�ˆ�1���Û���ÿÿÿÿÿÿÿÿ;���ˆ�ÿÿÿÿ#Ž�Charles Hermite9������H�Á�'��� €$�€���˜˜B˜€ �‚ÿ�Charles Hermite��b���;���ˆ�#Ž�'��� €v�€���¦€�€�‚ÿ�Charles Hermite (1822-1901) was a famous mathematician.���U���$���Á�xŽ�1���Î���ÿÿÿÿÿÿÿÿ<���xŽ�ÿÿÿÿñŽ�Recall the Bernstein Basis FunctionsG������#Ž�¿Ž�*��� $€:�€���˜˜B˜€ �€�‚ÿ�Bernstein Basis Functions���2������xŽ�ñŽ�.��� ,€ �€��P�œÈ€�†"€��š�‚ÿ����B������¿Ž�3�1���N��ÿÿÿÿÿÿÿÿ=���3�ÿÿÿÿ@À�Question 1 Answer4��� ���ñŽ�g�*��� $€�€���˜˜B¤€ �€�‚ÿ�Answer���Í���£���3�@À�*��� "€G�€���¤€�€�‚ÿ�They share the same direction, but the Hermite tangent�vector is n times the length of the first leg of the Bég�@À�ñŽ�zier�polygon, where n is the degree of the curve.���B������g�‚À�1���€��ÿÿÿÿÿÿÿÿ>���‚À�ÿÿÿÿÀÂ�Question 2 Answer4��� ���@À�¶À�*��� $€�€���˜˜B¤€ �€�‚ÿ�Answer���Ô���ª���‚À�ŠÁ�*��� "€U�€���¤€�‚‚‚‚ÿ�There are many ways, given that the end vector�of a given segment must match the start vector�of the next segment.��One way is to take the central difference, that is���6������¶À�ÀÁ�1��� 2€ �€��P�¤È€�†"€��›�€�‚ÿ�����'������ŠÁ�çÁ�$��� €�€���¤€�‚ÿ���Y���/���ÀÁ�@Â�*��� $€^�€���¦€�€ �€�‚ÿ�where �a� is an appropriate scalar constant.��€���W���çÁ�ÀÂ�)��� "€®�€���¤€�‚€�‚ÿ��Special care must be taken with the first and last�vectors of the set of segments.���B������@Â�Ã�1���ó��ÿÿÿÿÿÿÿÿ?���Ã�ÿÿÿÿ³Å�Question 3 Answer4��� ���ÀÂ�6Ã�*��� $€�€���˜˜B¤€ �€�‚ÿ�Answer���h���C���Ã�žÃ�%��� €†�€���¤€�‚‚ÿ�The derivative of the Bézier curve at the endpoints is given by���2������6Ã�ÐÃ�.��� ,€ �€��P�¤È€�†"€��œ�‚ÿ����\���6���žÃ�,Ä�&��� €l�€���¤€�‚‚‚ÿ��The intermediate points can easily be determined:���2������ÐÃ�^Ä�.��� ,€ �€��P�¤È€�†"€���‚ÿ����S���-���,Ä�±Ä�&��� €Z�€���¤€�‚‚‚ÿ��This gives the Bézier form of the curve:���2������^Ä�ãÄ�.��� ,€ �€��P�¤È€�†"€��ž�‚ÿ����N���(���±Ä�1Å�&��� €P�€���¤€�‚‚‚ÿ��To get the Hermite form, substitute���2������ãÄ�cÅ�.��� ,€ �€��P�¤È€�†"€��Ÿ�‚ÿ����P���+���1Å�³Å�%��� €V�€���¤€�‚‚ÿ��This gives the Hermite basis functions.��B������cÅ�õÅ�1���T��ÿÿÿÿÿÿÿÿ@���õÅ�ÿÿÿÿÈ�Question 4 Answer4��� ���³Å�)Æ�*��� $€�€���˜˜B˜€ �€�‚ÿ�Answer���A������õÅ�jÆ�$��� €:�€���¤€�‚ÿ�The Hermite form possesses��H������)Æ�²Æ�,��� (€8�€��2˜¦¨‚l€ �€�ƒ‚ÿ�·��endpoint interpolation��D������jÆ�öÆ�-��� *€.�€��r˜ì¨<‚l€ �€�ƒ‚ÿ�·��affine invariance����Ã���²Æ�È�N��� j€‡�€��r�˜¤¨‚€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ�This property arises trivially, since there are only two points�in the Hermite equation, �p�0� and �p�1�. �m�0� and �m�1� are vectors�determined by the points and are not transformed directly.��;��� ���öÆ�BÈ�1���+B��ÿ �üƒ�A���BÈ�È�ÈÇ�Blossoming?������È�È�*��� $€*�€���˜˜B¤€ �€�‚ÿ�Topic 5: Blossoms���F��� ���BÈ�ÇÈ�&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn����Î���È�ÚÉ�E��� X€�€��2˜¤ª‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��what a blossom is, and the connection between blossoms and CAGD,��·��the characteristics of blossoms,��·��the application of blossoming to Bézier and B-spline curves,��·��the uses of blossoms in CAGD.��'������ÇÈ�Ê�$��� €�€���¤€�‚ÿ���2������ÚÉ�3Ê�.��� ,€ �€��0¤¨€�†"€���‚ÿ����?������Ê�rÊ�,��� (€&�€���¤€�‚‚€ �€�‚ÿ����Introduction���*������3Ê�œÊ�'��� €�€��0¤ª‚l€�‚ÿ���~���W���rÊ�Ë�'��� €®�€��0�‘€»€€�‚ÿ�What's in a name? That which we call a rose, by any other name would smell as sweet.��*������œÊ�DË�'��� €�€��0¤ª‚$€�‚ÿ���S���#���Ë�—Ë�0��� 0€F�€��0¤ª‚Å…€�ƒ€ �€�‚ÿ��Shakespeare, �Romeo and Juliet���*������DË�ÁË�'��� €�€��0¤ª‚l€�‚ÿ���2�� ��—Ë�óÍ�&��� €�€��0�¤ª€�‚ÿ�In some instances, the name given to an object has little importance. A fragrance, for example, does not depend on its name. This is not true in mathematics, however; labels are significant. Well-chosen notation serves not only as a tag, but also suggests how a concept is defined and used. This is especially true of a technique called blossoming. Its power lies in its ability to suggest fundamentals, algorithms, and theorems in CAGD through labeling. The theory of blossoms was introduced by Ramshaw and de Casteljau.��'������ÁË�Î�$��� €�€���¤€�‚ÿ���2������óÍ�LÎ�.��� ,€ �€��0¤¨€�†"€���‚ÿ������-��Î�_�Ú��� ‚[�€���¤€�‚‚€ �€�‚€�‚âÙZ…€�‰€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€ �€�€�€�€ �€�€�€�€�€�€�€�€ �€�€�€�€ �€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ����Blossom Basics����The basic principle of blossoming arises from ��linear interpolation.�� Recall that �b�(0) and �b�(1) are points on the line segment at t = 0 and t = 1. Any point on the line is given by �b�(�a�).LÎ�_�È� The distance from �b�(0) to �b�(�a�) is proportional to �½� 0 - �a� �½�. Similarly, the distance from �b�(�a�) to �b�(1) is proportional to �½� 1 - �a� �½�. It is very useful to think of �a� as a measure of how far a point travels from �b�(0) to �b�(�a�). For example, if �a� = 0.5, then �b�(�a�) is halfway between �b�(�a�) and �b�(1).��•����LÎ�ô���� Ѐ+�€���¤€�‚€�€�€�€�€�€�‚‚È2�ExecProgram("demo1.exe Blossom_Interpolation", 0)�€�‡"€�� �‰€�‚ÿ��The following interactive demonstration is similar to the introductory linear interpolation demonstration shown in Topic 2, "Preliminary Mathematics." However, the demonstration here emphasizes the affine transformation between the linear space of �a� and the line segment �b�(0)-�b�(1).�������Click on the graphic for a demonstration of linear interpolation, with a display of the affine variable.��Use the left mouse button to drag the control points or the interpolated point, and observe the behavior of the affine variable.��¯����_�£�”��� ö€7�€���¤€�‚‚€ �€�‚€�‚€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�â™|ò*€�‰€�€�€�€�€�‚‚€�€�‚ÿ����Notation����To further emphasize this relationship between �a� and �b�(0)-�b�(1), the functional notation is dropped and simply written �b�(0) = �0�, �b�(1) = �1�, and �b�(�a�) = �a�. �a� is spoken of loosely as the ��affine distance�� from �0� to �a�.��Essentially, a point is designated by its parameter value �a� and the endpoints of the segment on which it lies, with the understanding that the point is obtained by linear interpolation along the line. This is shown in the following figure, which clarifies the new nomenclature:����º��ô�¿�b��� ’€y�€���¤€�‚‡"€��¡�‚‚€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��¢�‚‚‚ÿ�����Next an important extension is made to the notation in order to describe two lines that share a point. This is done by adding an additional digit, as shown in the following figures. First, add the digits to the two individual lines. The first line is identified with a leading �0�; it goes from �00� to �01�. The second line has a leading �1�, going from �10� to �11�.�����Next, bring the lines together so that they share an endpoint:��Ñ��C��£� �Ž��� ꀋ�€���¤€�‚‡"€��£�‚‚€�€�€�€�‚‚€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��¤�‚‚‚€ �€�‚€�‚€�€�€�€�‚ÿ�����Note here (this is most important, even though it seems obvious) that point �01� and point �10� are identical.��When there was a single line, the point �a� was on the line from �0� to �1�. In the two-line figure, if the point �0a� is on the line from �00� to �01�, then where is the point �1a�? Clearly, on the new line:�������Point Ordering Is Not Significant����The next generalization comes from noticing that the order of the digits is not important: �0a� or �a0� can be used to identify uniquely the point on the first line. Therefore, the digits may be in any order:��'������¿�· �$��� €�€���¤€�‚ÿ���ˆ���.��� �? �Z��� „€\�€��P�¤È€�€�€ �€�€�€�‚€�€�€ �€�€�€�‚€�€�€ �€�€�€�‚ÿ�01� �º� �10�,��a0� �º� �0a�,��a1� �º� �1a�.��P��á��· ��o��� ¬€Å�€���¤€�‚€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��¥�‚‚€�€�€�€�€�€�€�€�‚ÿ��Now consider a new line from �0a� to �1a�. Where is �aa�? It is the affine distance �a� between the line's endpoints, �0a� and �1a�. This is shown in the following figure:�����Furthermore it should be clear that �aa�, by its construction, is a point on a quadratic Bézier curve with control points �00�, �01�, and �11�. This follows from understanding de Casteljau's algorithm, which is based on repeated linear interpolation. The Bézier curve is shown in the following figure:����|��? �«@�”��� ö€ý�€���¤€�‚‡"€��¦�‚‚‚‚‡"€��§�‚‚€�€�€�€�€�€�âÊ8Y €�‰€�‚‚€�€�€�€�€ �€�€ �€�€�€�€�€�€�€�€�€�€�€�‚ÿ�����To add another control point, add another digit.�����Where are the points �00a�, �0a1�, and �a11�? ��Answer.����Here a convention is adopted of ordering the digits from smallest to largest. For example, �0a1� is pref�«@�È�erred to �1a0�, so long as 0 �£� a �£� 1. Sometimes commas are used to separate the digits, if required for clarity. The point �0�,�.5�,�1� is not �0.5� �1�.��)��–���ÔB�“��� ô€/�€���¤€�‚€�€�âîÄ@€€�‰€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��¨�‚‚€�€�€�€�€�€�€�€�€�€�‚ÿ��In the last figure, where is the point �aaa�? It can be found by recursive replacement. This is the so-called ��blossoming principle.�� First, find �00a�, �0a1�, and �a11� on the appropriate lines. Then, find �0aa� and �aa1�. Finally, �aaa� may be found on the line between �0aa� and �aa1�:�����By construction, the point �aaa� is on the Bézier curve with control points �000�, �001�, �011�, and �111�:��K������«@�C�8��� @€(�€���¤€�‚‡"€��©�‚‚‚€ �€�‚‚ÿ�������Exercise����’���Y���ÔB�±C�9��� B€²�€��¤‚l€�ƒ€�€�ââZËû€�‰€�‚ÿ�1.�Draw five points and find the point �aaaa� for a degree 4 Bézier curve.� �Answer.����õ���Å���C�¦D�0��� .€‹�€���¤€�‚‚€ �€�‚‚‚‚ÿ����Summary of Blossoming from Linear Interpolation���The generalization of this method for a Bézier curve of any degree should now be clear. The principles generated thus far may be summarized:���Î��L��±C�tF�‚��� Ò€™�€��T˜î:‚l€�ƒ€�€�€�€�‚ƒ€�€�€ �€�€�€�€ �€�€�€�‚ƒ€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ�1.�There is a Bézier curve of degree n that is given by points of the form �aa....a�. There are n �a�'s.�2.�The points are equivalent regardless of the ordering of the digits: �a10� �º� �01a� �º� �0a1�.�3.�The point �0...0�a�1...1� is an affine distance �a� along the line from �0...0�0�1...1� to �0...0�1�1...1�. In other words:��i���(���¦D�ÝF�A��� R€P�€���˜µ€€�€�€�€�€�€�€�€�€�€�‚ÿ�0a1� = (1 - �a�) �001� + (�a�) �011�.��L���'���tF�)G�%��� €N�€���˜ì€�‚ÿ�This is simple linear interpolation.��'������ÝF�PG�$��� €�€���¤€�‚ÿ���2������)G�‚G�.��� ,€ �€��0¤¨€�†"€���‚ÿ����m��#��PG�ïI�J��� b€G�€���¤€�‚‚€ �€�‚€�‚€�€�€�€�€�€�€�€�‚ÿ����The History of Blossoms����It was proved by Ramshaw and de Casteljau that for every polynomial P(u) of degree n, there is a unique function of n variables p(u�1�, u�2�, ..., u�n�) that has the three properties listed in the summary above. Ramshaw called this function the blossom of the polynomial P(u). Quite often this blossom is given only as a list of its variables, as shown. The blossom notation suggests methods and algorithms naturally. The de Casteljau algorithm arose by recursively applying the principles until �a...a� was found.��'������‚G�J�$��� €�€���¤€�‚ÿ���2������ïI�HJ�.��� ,€ �€��0¤¨€�†"€���‚ÿ����ã���ª���J�+K�9��� @€U�€���¤€�‚‚€ �€�‚€�‚€ �€�‚‚ÿ����The Formal Definition of Blossoms����To define a blossom, start with P(u), a polynomial of degree d. For an integer N �³� d, the Nth blossom of the polynomial P is���2������HJ�]K�.��� ,€ �€��P�¤È€�†"€��ª�‚ÿ����'������+K�„K�$��� €�€���¤€�‚ÿ���©��B��]K�-M�g��� œ€‡�€��T˜ì:„l¡…€�ƒƒ€�€�€ �€�‚ƒ†"€��«�ƒ€�€�€ �€�‚ƒƒ€�€�€ �€�‚ÿ�1.�P(u) = p(u, u, ..., u) �(5.1)����This property is known as �diagonal agreement�.�2.���(5.2)����This property is one of �symmetry�. The u values may be interchanged without modifying the blossom.�3.�p(..., as + bt, ...) = a p(..., s, ...) + b p(..., t, ...) if a + b = 1�(5.3)����This is the �multiaffinity� property.��M������„K�zM�0��� 0€:�€���¤€�‚‚€ �€�‚€�‚ÿ����Examples of Blossoms�����W��� ���-M�ÑM�M��� j€�€��¤‚l€�†"€��¬�‚†"€��­�‚†"€��®�‚†"€��¯�‚ÿ����������Ö���¯���zM�§N�'��� €_�€���¤€�‚‚‚ÿ��For any polynomial P and integer N, there exists one and only one blossom. Generally speaking, however, the blossom properties are more important than the blossom itself.���2������ÑM�ÙN�.��� ,€ �€��0¤¨€�†"€���‚ÿ����B������§N�O�0��� 0€$�€���¤€�‚‚€ �€�‚€�‚ÿ����Exercises�����^���6���ÙN�yO�(��� €l�€��˜¤‚l€�ƒ‚ÿ�2.�Deduce the blossoms for the following functions:��{���$���O� €�W��� ~€L�€��T˜¤È‚´€�ƒ†"€��°�ƒâµÀxX€�‰€�‚ƒ†"€��±�ƒâAÑ€�‰€�‚ÿ�A.�����Answer.���B.�����Answer.����������������yO� €�È���¡���yO��f��� š€C�€��T˜ì:‚l€�ƒ€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�âWÈrö€�‰€�‚ÿ�3.�Do other blossom values have meaning apart from �a...a�, �0...01...1�, etc.? For example, what is the meaning of �1�,�2�,�3� or �2.5�,�6�,�10�? ��Answer.����'������ €�:�$��� €�€���¤€�‚ÿ���2�������l�.��� ,€ �€��0¤¨€�†"€���‚ÿ����@�� ��:�¬ƒ�4��� 6€�€���¤€�‚‚€ �€�‚€�‚‚‚‚ÿ����The Power of the Blossom Form����Blossoms are very powerful. Once the principle is understood, methods may be generated to evaluate the Bézier curve, prove properties, define and evaluate B-spline curves, convert between B-spline and Bézier form, and much more.��The following two sections are intended for those who want more practice with blossoms. The material proves some new and some old things about Bézier curves by using blossoms. For example, one thing shown is that Bézier control points always have the form��'������l�Óƒ�$��� €�€���¤€�‚ÿ���2������¬ƒ�„�.��� ,€ �€��P�¤È€�†"€��²�‚ÿ����f���4���Óƒ�k„�2��� 4€h�€���¤€�‚ã {œ€�‰€�‚‚ÿ��To see the background material, ��click here.�����2������„�„�.��� ,€ �€��0¤¨€�†"€���‚ÿ����;��Õ��k„�؇�f��� š€«�€���¤€�‚‚€ �€�‚€�‚€�€�€�€�€�€�‚‚ã¾Öó]€�‰€�‚‚€�€�€�€�€�€�‚ÿ����Evaluating a Blossom from Bézier Control Points����Suppose that a set of Bézier control points is given. What is the point given by the blossom at �u�1�...�u�n�? The blossom properties lead to an algorithm that evaluates the blossom. It has the familiar form of iterated linear interpolation.��To see the development of the algorithm, ��click here.����As with the de Casteljau algorithm, let the control points be the base of a systolic array. Then generate the points, row by row, in the array using linear interpolation. The final value is the blossom value. The difference in this algorithm is that the linear interpolation value changes for each row. It is �u�1� in the first row, �u�2� in the second, and so forth:��'������„�ÿ‡�$��� €�€���¤€�‚ÿ���2������؇�1ˆ�.��� ,€ �€��P�œÐ€�†"€��³�‚ÿ����*������ÿ‡�[ˆ�'��� €�€��0¤ª‚l€�‚ÿ���2������1ˆ�ˆ�.��� ,€ �€��0¤¨€�†"€���‚ÿ����·��ƒ��[ˆ�DŠ�4��� 6€�€��0¤ª‚l€�‚‚€ �‚€�‚‚‚‚ÿ����EvalBlossomProg: A General Blossom Algorithm���The de Casteljau algorithm generalizes to the EvalBlossom procedure. This assumes a [0, 1] range of curve parameterization, which may be a limitation in some circumstances. If EvalBlossom is extended to handle an arbitrary blossom parameterization, the result is the generalization known as EvalBlossomProg.��The algorithm is given by��'������ˆ�kŠ�$��� €�€���œ€�‚ÿ���ª���w���DŠ�‹�3��� 6€î�€��P�œÈ€�€�€�€�€�‚‚‚ÿ�EvalBlossomProg  (�b�[i], . . ., �b�[i + d], u[1], . . ., u[d], t[i], . . ., t[i + 2d - 1])��for k = 0 to (d - 1) do��D������kŠ�Y‹�&��� €<�€��P�œ‘€€�‚ÿ�for j = 0 to (d - k - 1) do��¶���|���‹�Œ�:��� D€ø�€��P�œÙ€€�‚‚€�€�€�€�€�€�‚ÿ�Beta = (u[k + 1] - t[i + k + j]) / (t[i + d + j] - t[i + k + j])�Alpha = 1 - Beta��b�[j] = Alpha �b�[j] + Beta �b�[j + 1]��,������Y‹�;Œ�&��� € �€��P�œ‘€€�‚ÿ�end��@������Œ�{Œ�,��� (€(�€��P�œÈ€�‚€�€�‚ÿ�end�return �b�[0]��Ñ���¥���;Œ�L�,��� &€K�€���œ€�‚€�€�‚ÿ��In this algorithm �b�[i] are the de Boor points, which will be discussed in depth in Topic 6, "The B-Spline Curve." t[i] are the blossom parameterization values.��*������{Œ�v�'��� €�€��0¤ª‚l€�‚ÿ���2������L�¨�.��� ,€ �€��0¤¨€�†"€���‚ÿ����Ç��:��v�{Á���� è€w�€���¤€�‚‚€ �€�‚€�‚‚‚È*�ExecProgram("demo3.exe Cubic_Blossom", 0)�€�‡"€��´�‰€�€�€�‚‚€�€�â çÄÄ€�‰€�‚ÿ����Demonstration of the Cubic Blossom����The following interactive demonstration allows one to vary the blossom arguments and see the resulting value computed by EvalBlossomProg.�������Click on the graphic for a demonstration of the cubic blossom.��Use the left mouse button to drag the control points or the blossom �a� values, and observe the behavior of the blossom point.��The question arises: Why determine any blossom value that is not on the curve, th¨�{Á�È�at is, not �aaa�? One example comes from the ��reparameterization�� of the curve. Another way to reparameterize the curve is to use EvalBlossomProg. A Bézier curve is conventionally parameterized with a parameter in the range from 0 to 1. Now, what if another segment of the curve with parameter values from 2 to 3 were sought in terms of the Bézier control points?��ü��u��¨�wÄ�‡��� Ü€í�€���¤€�‚€�€�€�€�€�€�€�€�‚‚‚‚È*�ExecProgram("demo3.exe Param_Blossom", 0)�€�‡"€��µ�‰€�€�€�‚ÿ��For the cubic Bézier curve, the new control points will have the form �222�, �223�, �233�, and �333�. The first and last points are, of course, on the curve and are derived from de Casteljau's algorithm, but the others require the more general EvalBlossomProg.��The following interactive demonstration permits the selection of a parameter region from -2 to +2, and the control points will be computed by EvalBlossomProg.������� Click on the graphic for a demonstration of the reparameterized blossom.��Drag the parameter sliders to vary the �u� values, or move the control points of the segment parameterized between 0 and 1.��Ð���©���{Á�GÅ�'��� €S�€���¤€�‚‚‚ÿ��More importantly than reparameterization, blossoming gives the facility to define a new curve: the B-spline. This is fully covered in Topic 6, "The B-Spline Curve."���2������wÄ�yÅ�.��� ,€ �€��0¤¨€�†"€���‚ÿ����0��ý���GÅ�©Æ�3��� 4€û�€���¤€�‚‚€ �€�‚€�‚‚‚ÿ����What Has Been Accomplished in This Topic����A strong base of knowledge concerning the blossom form has been built. This has allowed the investigation of the Bézier curve more deeply and will motivate the development of B-spline curves in Topic 6.���2������yÅ�ÛÆ�.��� ,€ �€��0¤¨€�†"€���‚ÿ����)������©Æ�Ç�%��� €�€���¤€�‚‚ÿ����Ä���z���ÛÆ�ÈÇ�J��� d€ô�€��0¤¨‚l㤓h€�‰€�‚ã!}¤÷€�‰€�‚ã 8ûU€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: The B-Spline Curve�����Go to the previous topic: Interpolation����O������Ç�È�1���ñ��ÿÿÿÿÿÿÿÿB���È�ÿÿÿÿ¹É�Recall Linear Interpolation...F������ÈÇ�]È�'��� €>�€���˜˜B˜€ �‚ÿ�Recall Linear Interpolation:��ˆ���a���È�åÈ�'��� €Â�€��0�¦¨€�‚‚ÿ�Given two points in space, a line in parametric form can�be defined that passes through them:���3������]È�É�/��� .€ �€��p�¦¨È€�†"€��¶�‚ÿ����¡���i���åÈ�¹É�8��� @€Ô�€��0�¦¨€�‚†"€��·�‚‚€�€�‚ÿ�����Thus �l�(t) is a point somewhere on the line between the�two points, depending on the parameter t.��@������É�ùÉ�1���F��ÿÿÿÿÿÿÿÿC���ùÉ�ÿÿÿÿÿÊ�Affine Distance9������¹É�2Ê�'��� €$�€���˜˜B¤€ �‚ÿ�Affine Distance��›���c���ùÉ�ÍÊ�8��� @€Æ�€���¤€�€�€�€�€�€�€�‚‚ÿ�The affine distance of a point �a� to a point �b��on a line �bc� is the ratio of the distances:���2������2Ê�ÿÊ�.��� ,€ �€��P�¤È€�†"€��¸�‚ÿ����B������ÍÊ�AË�1���Û���ÿÿÿÿÿÿÿÿD���AË�ÿÿÿÿÚË�00a, 0a1, and a11;������ÿÊ�|Ë�'��� €(�€���˜˜B˜€ �‚ÿ�00a, 0a1, and a11��^���,���AË�ÚË�2��� 4€Z�€���œ€�‚‚†"€��¹�€�‚ÿ�The answer is best shown graphically:������E������|Ë�Ì�1���ñ��ÿÿÿÿÿÿÿÿE���Ì�ÿÿÿÿËÍ�Blossoming Principle>������ÚË�]Ì�'��� €.�€���˜˜B¤€ �‚ÿ�Blossoming Principle��<��ÿ���Ì�™Í�=��� H€ÿ�€���¤€�‚‚€�€�€�€�€�€�‚‚ÿ�Informally, the digit that differs between two blossoms is�replaced with another value, giving the affine distance�along the line segment for the new value.��Thus, �axx� is found on the line segment between �bxx� and��cxx�, an affine distance given by���2������]Ì�ËÍ�.��� ,€ �€��P�¤È€�†"€��º�‚ÿ����G������™Í�Î�1���H��ÿÿÿÿÿÿÿÿF���Î�ÿÿÿÿÏ�Answer to Question 1 A0��� ���ËÍ�BÎ�'��� €�€���˜˜B¤€ �‚ÿ�Answer��7������Î�yÎ�%��� €$�€���¦€�‚‚ÿ�The blossom of���2������BÎ�«Î�.��� ,€ �€��P�¦È€�†"€��»�‚ÿ����6������yÎ�áÎ�&��� € �€���¦€�‚‚‚ÿ��is given by���2������«Î�Ï�.��� ,€ �€��P�¦Æ€�†"€��¼�‚ÿ����G������áÎ�ZÏ�1���H��ÿÿÿÿÿÿÿÿG���ZÏ�ÿÿÿÿt��Answer to Question 1 B0��� ���Ï�ŠÏ�'��� €�€���˜˜B˜€ �‚ÿ�Answer��7������ZÏ�ÁÏ�%��� €$�€���¦€�‚‚ÿ�The blossom of���2������ŠÏ� ��.��� ,€ �€��P�¦È€�†"€��½�‚ÿ�����������������ÁÏ� ��Ï�6������ÁÏ�B��&��� € �€���¦€�‚‚‚ÿ��is given by���2������ ��t��.��� ,€ �€��P�¦Æ€�†"€��¾�‚ÿ����E������B��¹��1���Â���ÿÿÿÿÿÿÿÿH���¹��ÿÿÿÿ6�Answer to Question 20��� ���t��é��'��� €�€���˜˜B˜€ �‚ÿ�Answer��M���&���¹��6�'��� €L�€���œ€�€�‚ÿ�See Topic 6, "The B-Spline Curve."���L������é��‚�1���— ��ÿÿÿÿÿÿÿÿI���‚�Ê�0�Blossom Background MaterialH���!���6�Ê�'��� €B�€���˜˜B˜€ �‚ÿ�Background of the Blossom Form��K���'���‚��$��� €N�€���œ€�‚ÿ�Uses of blossoms are described here:��Ž���Y���Ê�£�5��� :€²�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚ÿ�·��the Bézier curve in blossom form��·��subdivision of the Bézier curve using blossoms��'�������Ê�$��� €�€���¤€�‚ÿ���2������£�ü�.��� ,€ �€��0¤¨€�†"€���‚ÿ����)������Ê�%�%��� €�€���¤€�‚‚ÿ����¾���…���ü�ã�9��� @€ �€���œ€ �€�‚€�‚‚‚†"€��¿�‚‚ÿ�The Bézier Curve in Blossom Form����Using the three properties of blossoms previously described, the following can be observed������2������%��.��� ,€ �€��P�œÈ€�†"€��À�‚ÿ����H����ã�]�8��� >€!�€���œ€�‚‚‚‚‚‚‚âÊÚói€�‰€�‚ÿ��The point p(u, u,..., u) is on the curve P(u) by the diagonal property. In the first equation, the first u in the argument list is written as an affine combination: u = (1 - u).0 + u.1.��In the second line, the polynomial is factored by multiaffinity.��These steps are repeated for the next u in the argument list, and the terms are combined using the symmetry property to arrive at the third line. Continuing in this manner, the argument list is exhausted.��The polynomial, as seen in the last line, is a summation of coefficients, written as blossoms, multiplied by the ��Bernstein basis polynomials.�� This polynomial is a Bézier curve of degree N. This can already be seen to be emerging in the third equation; the polynomials in each term are quadratic Bernstein polynomials.��Ð���©����-�'��� €S�€���œ€�‚‚‚ÿ��The coefficients, or control points, are blossoms evaluated with only zeros and ones, so the blossoms p(0, 0,..., 1, 1) are then the Bézier control points, that is,���2������]�_�.��� ,€ �€��P�œÈ€�†"€��Á�‚ÿ����®���ˆ���-� �&��� €�€���œ€�‚‚ÿ��There are i ones in the blossom. This control point is the blossom evaluated at (N - i) zeros and i ones; the order does not matter.��'������_�4 �$��� €�€���¤€�‚ÿ���2������ �f �.��� ,€ �€��0¤¨€�†"€���‚ÿ����)������4 � �%��� €�€���¤€�‚‚ÿ����_���4���f �î �+��� &€h�€���œ€ �€�‚€�‚ÿ�Subdivision of the Bézier Curve Using Blossoms�����É���œ��� �· �-��� (€9�€��T˜î2‚l€ �€�ƒ‚ÿ�·��To subdivide a Bézier curve using blossoms, first let [s,t] be an arbitrary interval. Now, what are the Bézier control points over the interval [s,t]?��q���E���î �( �,��� (€Š�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Write the point on the curve at which subdivision is desired as��5������· �] �1��� 2€ �€��T˜îÆ‚l€�†"€��Â�‚ÿ����g���@���( �Ä �'��� €€�€��˜î‚l€�‚ÿ�The diagonal property ensures that the point is on the curve.��—���k���] �[ �,��� (€Ö�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Proceeding as before with the Bézier blossom description, with zeros becoming s, and ones becoming t,��?��� ���Ä �š �5��� :€�€��T˜îÆ„l¡…€�†"€��Ã�ƒ‚ÿ���(5.4)��ˆ���a���[ �" �'��� €Â�€��˜î‚l€�‚ÿ�There are i t's in this blossom. This is the ith Bézier control point over the interval [s,t].��'������š �I �$��� €�€���œ€�‚ÿ���A����" �Š�'��� €5�€��T�˜¤‚€�‚ÿ�Hence, if there were a method to evaluate the blossom of a polynomial, it could easily reparameterize over any interval. Specifically, it could be reparameterized over the interval [s,t] = [0,a] or [a,1]. This is equivalent to subdividing the Bézier curve at a parameter value a.��'������I �±�$��� €�€���œ€�‚ÿ���2������Š�ã�.��� ,€ �€��0¤¨€�†"€���‚ÿ����M������±�0�2��� 4€6�€���œ€�‚‚ãë·õ§€�‰€�‚ÿ�����Return to Blossoms����L������ã�|�1���?��ÿÿÿÿÿÿÿÿJ���|�ÿÿÿÿd@�Bernstein Basis PolynomialsI������0�Å�*��� $€>�€���˜˜B˜€ �€�‚ÿ�Bernstein Basis Polynomials���a���<���|�2@�%��� €x�€���œ€�‚‚ÿ�Recall that the BernsÅ�2@�0�tein basis polynomials are given by���2������Å�d@�.��� ,€ �€��P�œÈ€�†"€��Ä�‚ÿ����J������2@�®@�1���÷��ÿÿÿÿÿÿÿÿK���®@�ñ@�·E�The EvalBlossom AlgorithmC������d@�ñ@�'��� €8�€���˜˜B˜€ �‚ÿ�The EvalBlossom Algorithm��9������®@�*A�&��� €&�€��T�˜¤‚€�‚ÿ�Use the property��4������ñ@�^A�0��� 0€ �€��˜î‚l€�†"€��Å�‚ÿ����‰��c��*A�çB�&��� €Ç�€���¤€�‚‚ÿ�Given the Bézier control coefficients (in blossom form) and a value for u, recursion is performed, in a similar way to de Casteljau's method. In this iterative procedure, the single element on the bottom row is the desired blossom value. The array provides a schematic for evaluating a blossom. The formal algorithm, called EvalBlossom, is as follows:���{���H���^A�bC�3��� 6€�€��P�œÈ€�€�€�€�€�‚‚‚ÿ�EvalBlossom (�b�[0], ..., �b�[d], u[1], ..., u[d])��for i = 1 to d do��@������çB�¢C�&��� €4�€��P�œ‘€€�‚ÿ�for j = 0 to (d - i) do��c���.���bC�D�5��� :€\�€��P�œÙ€€�€�€�€�€�€�‚ÿ�b�[j] = (1 - u[i]) �b�[j] + u[i] �b�[j + 1]��,������¢C�1D�&��� € �€��P�œ‘€€�‚ÿ�end��@������D�qD�,��� (€(�€��P�œÈ€�‚€�€�‚ÿ�end�return �b�[0]��Ç��� ���1D�8E�'��� €A�€���œ€�‚‚‚ÿ��If u = u[0] = u[1] = . . . = u[N], then EvalBlossom is exactly the same as the de Casteljau algorithm for evaluating a Bézier curve at a parameter value u.���2������qD�jE�.��� ,€ �€��0¤¨€�†"€���‚ÿ����M������8E�·E�2��� 4€6�€���œ€�‚‚ãÑ3ÂF€�‰€�‚ÿ�����Return to Blossoms����C������jE�úE�1���Œ��ÿÿÿÿÿÿÿÿL���úE�ÿÿÿÿCG�Reparameterization@������·E�:F�*��� $€,�€���˜˜B˜€ �€�‚ÿ�Reparameterization��� ��à���úE�CG�)��� €Á�€���œ€�‚ÿ�While it is convenient to parameterize curves in�the interval [0, 1], there is nothing unique about�this range. The same curve may be parameterized�in the interval [-2, 2] if the blossom values are�selected appropriately.��C������:F�†G�1��� ^��wƒ �]�M���†G�ËG�zÃ�The B-Spline CurveE������CG�ËG�'��� €<�€���˜˜B¤€ �‚ÿ�Topic 6: The B-Spline Curve��F��� ���†G�H�&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn��Ù���œ���ËG�êH�=��� H€9�€��2˜¤ª‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��the characteristics of B-spline curves,��·��the derivation of B-spline curves from blossoming,��·��the differences between Bézier and B-spline curves.��*������H�I�'��� €�€��0¤ª‚l€�‚ÿ���2������êH�FI�.��� ,€ �€��0¤¨€�†"€���‚ÿ����,������I�rI�(��� €�€��0¤ª‚l€�‚‚ÿ����7������FI�©I�'��� € �€���¤€ �€�‚ÿ�Introduction���×��{��rI�€K�\��� †€÷�€��0¤ª‚l€�‚‚‚€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚‚ÿ��The discussion of B-spline curves begins by considering a Bézier curve in blossom form.��In the following interactive demonstration, a Bézier curve has control points �222�, �223�, �233�, and �333� in the parameter interval 2 to 3. Also shown are the blossom points �012�, �123�, �234�, and �345�. Move the Bézier control points and observe the behavior of the other points.���Q��Ö���©I�ÑL�{��� Ä€¯�€���¤È,�ExecProgram("demo3.exe BSpline_Blossom", 0)�€�‡"€��Æ�‰€�€�€�€�€�€�€�€�€�‚ÿ�����Click on the graphic for a demonstration of the reparameterized Bézier curve.��Observe how a small movement of the Bézier control points causes a dramatic change in the points �012�, �123�, �234�, and �345�.��¡��"��€K�~€���� Ì€E�€��0¤ª‚l€�‚âj€k߀�‰€�€�€�€�€�€�€�€�€�â_06-€�‰€�â¹;8€�‰€�‚‚€�€�€�€�€�€�€�€�‚ÿ��The additional points are computed with the ��EvalBlossomProg�� routine, which was introduced in the last topic on blossoms. The new points �012�, �123�, �234�, and �345� are called the ��de Boor�� points. When a curve is given by these points (instead of, for example, the Bézier control points), it is called a ��B-spline curve.�� ��Superficially it seems that the Bézier form of the curve is better; it interpolates the endpoints, has endpoint tangents, and more closely approximates the control polygon. The advantage of B-spline curves comes when another blossom point is added at an arbitrary point in space: �456�. This point, when taken with �123�, �234�, and �345�, defines a nÑL�~€�CG�ew B-spline curve segment that runs from parameter value 3 to 4, as shown in the next interactive demonstration.��*������ÑL�¨€�'��� €�€��0¤ª‚l€�‚ÿ�����¦���~€�¬�^��� Š€O�€���¤È)�ExecProgram("demo3.exe Join_Splines", 0)�€�‡"€��Ç�‰€�‚ÿ�����Click on the graphic for a demonstration of multiple curve segments. Move the control points, and observe how the two curve segments maintain their continuity.��Ÿ��g��¨€�Kƒ�8��� >€Ï�€��0¤ª‚l€�‚âÎï‡á€�‰€�‚‚‚‚ÿ��Notice that the two curve segments join smoothly without special effort. This is the strength and beauty of B-spline curves. In the cubic case they automatically join with ��C2 continuity,�� so long as the curve is not degenerate.��In the following interactive demonstration, start with a cubic B-spline curve, and then include additional control points.���N��ê���¬�™„�d��� –€×�€���¤È-�ExecProgram("demo3.exe Cubic_Spline_Add", 0)�€�‡"€��È�‰€�‚ÿ�����Click on the graphic for a demonstration of the multiple-segment cubic B-spline curve.��With the left mouse button, drag the existing control points or add new points; with the right mouse button, delete the last control point.��½��^��Kƒ�V‡�_��� Œ€½�€��0¤ª‚l€�‚€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ��The power of B-spline curves is that one can create with ease a very complex curve that is smoothly connected. They are formed from sequential shifts: �012�, then �123�, then �234�. This is true for B-spline curves of any degree. Points �012�, . . . , �345� can be computed from EvalBlossomProg with �222�, . . . , �333� as input. De Boor point �456� does not come from these input points; rather, it was arbitrarily added. It actually belongs with a different curve, the second curve segment. EvalBlossomProg can also compute a point �456�; this is, of course, different from the point that was added.��g��5��™„�½ˆ�2��� 2€k�€��0¤ª‚l€�‚€�€�‚‚‚‚ÿ��What is needed is a method to evaluate the curves' Bézier control points, given the B-spline points. This is not hard when recalling the blossom properties; affine replacement still works, and �aaa� is still on the curve.��These ideas may be used to derive the de Boor algorithm for evaluating B-splines.���2������V‡�ïˆ�.��� ,€ �€��0¤¨€�†"€���‚ÿ����,������½ˆ�‰�(��� €�€��0¤ª‚l€�‚‚ÿ����@������ïˆ�[‰�'��� €2�€���¤€ �€�‚ÿ�The de Boor Algorithm���i��à��‰�Ä‹�‰��� à€Å�€��0¤ª‚l€�‚€�€�€�€�€�€�‚‚‡"€��É�‚‚€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��Ê�€�€�€�€�€�€�‚ÿ��Start with the original set of de Boor points and find the point �2.5�, �2.5�, �2.5�, which is in the curve between parameter values 2 and 3.���Here is the starting set of de Boor points.��Begin with the first leg of the curve (�012� to �123�). The digits �1� and �2� are common; �0� changes to �3� as we progress along the curve.���Where is �1�, �2�, �2.5� in this progression? It is linearly interpolated along the line; it is 2.5/3 in affine distance along the polygon leg.��»���b���[‰�Œ�Y��� ‚€Æ�€��0¤ª‚l€�‚‡"€��Ë�€�€�€�€�€�€�€�€�€�€�€�€�€�‚‚ÿ���Similarly, �2�,� �2.5�, �3� is found on the second leg, and �2.5�, �3�, �4� on the last leg.���‘��õ���Ä‹�Ž�œ��� ñ�€��0�¤ª€�‡"€��Ì�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚‚‡"€��Í�€�€�€�€�€�€�€�‚‚‡"€��Î�‚ÿ��The replacement is repeated to find �2�,� �2.5�, �2.5� (as �1� goes to �3�) and �2.5�,� �2.5�, �3� (as �2� goes to �3�).���The final step yields �2.5�,� �2.5�, �2.5�, a point on the curve.���This process may be described as a systolic array.��*������Œ�:Ž�'��� €�€��0¤ª‚l€�‚ÿ���p���8���Ž�ªŽ�8��� @€p�€��0�¤ª€�€�€�€�€�€�€�‚‚ÿ�The condition to derive the point �1�, �2�, �2.5� is���3������:Ž�ÝŽ�/��� .€ �€��p�¤ªÈ€�†"€��Ï�‚ÿ����(������ªŽ��%��� €�€��0�¤ª€�‚ÿ���¿���–���ÝŽ�Ä�)��� €-�€��0¤ª‚l€�‚‚ÿ�As an exercise, be sure you are able to derive the other blossom points. If a parameter t is used instead of 2.5, then the condition above becomes���5������� À�1��� 2€ �€��p¤ªÈ‚l€�†"€��Ð�‚ÿ�����������Ä� À�CG�{���R���Ä�‡À�)��� "€¤�€��0¤ª‚l€�‚‚‚ÿ��For a different interval, starting at the parameter value i, the condition is���5������ À�¼À�1��� 2€ �€��p¤ªÈ‚l€�†"€��Ñ�‚ÿ����N���%���‡À� Á�)��� "€J�€��0¤ª‚l€�‚‚‚ÿ��Finally, change the degree to n:���5������¼À�?Á�1��� 2€ �€��p¤ªÈ‚l€�†"€��Ò�‚ÿ����€���W��� Á�¿Á�)��� "€®�€��0¤ª‚l€�‚‚‚ÿ��This is for a single step in the array. The formal de Boor algorithm is written as���?��� ���?Á�þÁ�5��� :€�€��p¤ªÈ„l¡…€�†"€��Ó�ƒ‚ÿ���(6.1)��T����¿Á�RÃ�=��� H€1�€��0¤ª‚l€�‚†"€��Ô�‚‚‚‚€�€�‚‚ÿ�����The u values are the parameter intervals; n is the degree of the curve.��This equation is the well-known de Boor algorithm. The de Boor points �d� are associated with the blossom counterparts. The location of each succeeding level of points is made clear by the blossoms.���2������þÁ�„Ã�.��� ,€ �€��0¤¨€�†"€���‚ÿ����œ���k���RÃ� Ä�1��� 2€Ö�€��0¤ª‚l€�‚‚€ �‚€�‚‚ÿ����Summary of B-Splines in Blossom Form���The EvalBlossomProg procedure, in its more general form, gives��(��Þ���„Ã�HÅ�J��� b€½�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒâÔÆ0v€�‰€�‚€ �€�ƒ‚ÿ�·��a method to evaluate B-spline curves,��·��a reparameterization method for B-spline curves (this also permits ��subdivision�� of the curve),��·��a method to transform between B-spline curve segments and Bézier curves.��Í���£��� Ä�Æ�*��� "€G�€��0¤ª‚l€�‚‚‚ÿ��So far, only cubic B-spline curves have been discussed. In the following interactive demonstration, examine the properties of B-splines of degree 2, 3, and 4.���·��T��HÅ�ÌÇ�c��� ”€«�€���¤È*�ExecProgram("demo3.exe Multi_Splines", 0)�€�‡"€��Õ�‰€�‚ÿ�����Click on the graphic for a demonstration of the multiple-segment B-spline curve of varying degree.��With the left mouse button, drag the existing control points or add new ones; with the right mouse button, delete the last control point.��Change the degree of the curve between quadratic, cubic, and quartic with the "Degree" button.��~���U���Æ�JÈ�)��� "€ª�€��0¤ª‚l€�‚‚‚ÿ��The B-spline control points in blossom form reveal much about the curve segment:���x����ÌÇ�ÂÊ�]��� ˆ€7�€��T˜î:‚l€ �€�ƒ€�€�‚€ �€�ƒ€�€�€�€�€�€�‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��The number of digits equals the degree (�012� is a control point for a cubic curve).��·��There are sequential shifts of adjacent digits (�012�, �123�, �234�,...).��·��The last digit of the first point and the first digit of the last point give the valid parameterization interval for the curve segment. In the cubic example, this interval is in the range 2...3.��·��Adding one new point to the existing set gives another segment of the B-spline curve. This new segment typically joins the existing curve with degree n - 1 continuity.��*������JÈ�ìÊ�'��� €�€��0¤ª‚l€�‚ÿ���2������ÂÊ�Ë�.��� ,€ �€��0¤¨€�†"€���‚ÿ����u��8��ìÊ�“Í�=��� H€s�€��0¤ª‚l€�‚‚€ �‚€�‚‚‚‡"€��³�‚ÿ����EvalBlossomProg���Topic 5, "Blossoms," discussed how the de Casteljau algorithm generalizes to the EvalBlossom procedure. Does this analog hold for the de Boor algorithm? If de Boor's algorithm is evaluated with different parameter values at each level of iteration (exactly as in EvalBlossom), it can be shown that the appropriate blossom values are produced from the de Boor points. The generalization is known as EvalBlossomProg.���Recall this progression from the topic on blossoms. The top line of the figure contains the Bézier control points of the curve.��W���/���Ë�êÍ�(��� €^�€��0¤ª‚l€�‚‚ÿ��Also, recall that the algorithm is given by��'������“Í�Î�$��� €�€���œ€�‚ÿ���ª���w���êÍ�»Î�3��� 6€î�€��P�œÈ€�€�€�€�€�‚‚‚ÿ�EvalBlossomProg  (�b�[i], . . ., �b�[i + d], u[1], . . ., u[d], t[i], . . ., t[i + 2d - 1])��for k = 0 to (d - 1) do��D������Î�ÿÎ�&��� €<�€��P�œ‘€€�‚ÿ�for j = 0 to (d - k - 1) do��¶���|���»Î�µÏ�:��� D€ø�€��P�œÙ€€�‚‚€�€�€�€�€�€�‚ÿ�Beta = (u[k + 1] - t[i + k + j]) / (t[i + d + j] - t[i + k + j])�Alpha = 1 - Beta��b�[j] = Alpha �b�[j] + Beta �b�[j + 1]��,������ÿÎ�áÏ�&��� € �€��P�œ‘€€�‚ÿ�end��@������µÏ�-��,��� (€(�€��P�œáÏ�-��CG�È€�‚€�€�‚ÿ�end�return �b�[0]��Ð���˜���áÏ�ý��8��� >€1�€���œ€�‚€�€�â«Fvw€�‰€�‚ÿ��In this algorithm the �b�[i] are the de Boor points. The t[i] are the blossom parameterization values for the B-spline, commonly known as ��knots.������Ù���-����*��� "€³�€��0¤ª‚l€�‚‚‚ÿ��What can be done with this algorithm? Any blossom value can be generated from the de Boor points. In particular, the Bézier control points may be needed, so that B-spline curves may be converted to Bézier curves.���2������ý��2�.��� ,€ �€��0¤¨€�†"€���‚ÿ����A��������s�0��� 0€"�€��0¤ª‚l€�‚‚€ �‚€�‚ÿ����Exercises������¨���2��f��� š€Q�€��pìª:‚l€�ƒ€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�â“ß…¾€�‰€�‚ÿ�1.�Given the set of points {�012�, �123�, �234�, �345�}, construct the equivalent Bézier control points {�222�, �223�, �233�, �333�}. For the answer, ��click here.����*������s�«�'��� €�€��0¤ª‚l€�‚ÿ���þ���¼����©�B��� R€y�€��pìª:‚l€�ƒ€�€�€�€�â³|�€�‰€�‚ÿ�2.�Why were the points �000� and �111� not sought, to give a parameterization interval of 0 to 1? What significance to the B-spline do these points have? For the answer, ��click here.����*������«�Ó�'��� €�€��0¤ª‚l€�‚ÿ���2������©��.��� ,€ �€��0¤¨€�†"€���‚ÿ����H������Ó�M�.��� ,€4�€��0�¤ª€�‚‚€ �‚€�‚ÿ����Periodic B-Splines����x��1���Å�G��� \€e�€���¤€�‡"€��Ö�€ �€�€�€�€�€�€�€�‚‚ÿ��If the first n and last n control points are made to coincide, then the curve's endpoints will match and it will form a closed curve. This is called a �periodic B-spline�. As before, each segment connects with its neighbors with degree n - 1 continuity. In the figure, �d�0� and �d�6� are coincident.���2������M�÷�.��� ,€ �€��0¤¨€�†"€���‚ÿ����‚����Å�y �z��� €�€��0¤ª‚l€�‚‚€ �‚€�‚€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ����Nonuniform B-Splines���Thus far, the blossom arguments have been integers. There is no particular reason for this, apart from clarity. The blossom arguments may be any real numbers and all the previous methods will still work. For instance, a Bézier curve may be parameterized over the interval [2.5, 3]. Or, the B-spline control points may be given as {�0�,�1�,�2.5� �1�,�2.5�,�3� �2.5�,�3�,�4� �3�,�4�,�5�}. The change that occurs is similar to that in changing the parameterization in Lagrange interpolation.��H��Ä��÷�Á �„��� Ö€‹�€��0¤ª‚l€�‚€�€�€�€�€�€�€�€�‚‚È,�ExecProgram("demo3.exe Non_Uni_Blossom", 0)�€�‡"€��×�‰€�‚ÿ��The following interactive demonstration shows the original cubic B-spline curve with its de Boor points, using integer blossom values {�012�, �123�, �234�, �345�}. The overlay curve uses the same points but with a different parameterization.�������Click on the graphic for a demonstration of different parameterizations of the same set of points.��Click on the "Uniform / Nonuniform" button to change between the two parameterizations of the curve.��•��!��y �V�t��� ¶€E�€��0¤ª‚l€�‚âçÒF•€�‰€�‚‚‚‚È&�ExecProgram("demo3.exe Knot_Demo", 0)�€�‡"€��Ø�‰€�‚ÿ��The blossoming principle still determines where to put points. They differ from the integer case because of different arguments. A B-spline curve with unequally spaced sequences is said to be ��nonuniform.�� As with Lagrange interpolation, the arguments are called knots.��Change the knot sequence in the following interactive demonstration and observe the effects.�������Click on the graphic for a demonstration of varying knot sequences.��Use the left mouse button to move the knot positions to create a new parameterization for the curve.��*������Á �€�'��� €�€��0¤ª‚l€�‚ÿ���2������V�²�.��� ,€ �€��0¤¨€�†"€���‚ÿ����G������€�ù�0��� 0€.�€��0¤ª‚l€�‚‚€ �‚€�‚ÿ����Basis Functions�������\���²�z�%��� €¸�€���¤€�‚‚ÿ�A single B-spline curve segment is defined much like a Bézier curve. It looks like this:���=��� ���ù�·�3��� 6€�€��P¤È‚¡…€�†"€��Ù�ƒ‚ÿ���(6.2)����«��z�ÝA�o��� ¬€Y�€���¤€�‚€�€�‚‚‚‚È*�ExecProgram("demo3.exe B·�ÝA�CG�Spline_Basis", 0)�€�‡"€��Ú�‰€�‚ÿ��where the �d� points are the de Boor points, the N(t) are the basis functions, and n is the degree of the curve.��The basis functions used here are different from the Bernstein basis functions used by the Bézier curve.�������Click on the graphic to gain a deeper understanding of the B-spline basis functions.��Drag the knot points to modify the basis functions; click the "Degree" button to change the degree of the curve.��f��>��·�CD�(��� €}�€���¤€�‚‚‚‚ÿ��The basis functions for the B-spline curves are placed next to each other and overlap in a similar way to the control points. When they are drawn together, bell-shaped functions are generated that are zero outside a certain region of "support." It is also clear that they are simply translations of each other, for uniform knots.��It can be seen that (in the cubic case) for any parameter value t, only four basis functions are nonzero; thus, only four control points affect the curve at t. If a control point is moved, it influences only a limited portion of the curve.��'������ÝA�jD�$��� €�€���¤€�‚ÿ���â���¶���CD�LE�,��� &€m�€���œ€�€ �€�‚‚ÿ�This locality of influence is known as the �local support property�. In the same way as the Bernstein polynomials, the B-spline basis functions conform to the partition of unity:���2������jD�~E�.��� ,€ �€��P�œÈ€�†"€��Û�‚ÿ����ô���Î���LE�rF�&��� €�€���œ€�‚‚ÿ��This is used to prove that any point on the curve is a convex combination of the de Boor points; that is, it must be in the convex hull of the control points associated with the nonzero basis functions.��'������~E�™F�$��� €�€���¤€�‚ÿ���2������rF�ËF�.��� ,€ �€��0¤¨€�†"€���‚ÿ����'��ç��™F�òH�@��� N€Ï�€���¤€�‚‚€ �€�‚€�‚â³åb€�‰€�‚‚‚ÿ����General Basis Functions����The basis functions considered so far have been cubic or of lower degree. Schoenberg first introduced the B-spline in 1949. He defined the basis functions using ��integral convolution�� (the "B" in B-spline stands for "basis"). Higher degree basis functions are given by convolving multiple basis functions of one degree lower.��Linear basis functions are just "tents" as shown below. When convolved together, they make piecewise parabolic "bell" curves.��Ó��Ÿ��ËF�ÅJ�4��� 6€A�€���¤€�‚†"€��Ü�‚‚‚‚‚‚ÿ�����The tent basis function (which has a degree of one) is nonzero over two intervals, the parabola is nonzero over three intervals, and so forth. This gives the region of influence for different degree B-spline control points. Each convolution results in higher order continuity between segments of the basis function.��When the de Boor points are weighted by these basis functions, the B-spline curve results:���=��� ���òH�K�3��� 6€�€��P¤È‚¡…€�†"€��Ý�ƒ‚ÿ���(6.3)��ˆ���b���ÅJ�ŠK�&��� €Ä�€���¤€�‚‚‚ÿ��Instead of integrating to evaluate the basis functions, a recursive formula has been derived:���=��� ���K�ÇK�3��� 6€�€��P¤È‚¡…€�†"€��Þ�ƒ‚ÿ���(6.4)��0��� ���ŠK�÷K�&��� €�€���¤€�‚‚‚ÿ��where���1������ÇK�(L�-��� *€ �€���ì€�†"€��ß�‚ÿ����*����÷K�RM�)��� €�€���¤€�‚‚‚‚‚ÿ��The terms in u represent the knot sequence, the spans over which the de Boor points influence the B-spline.��This recursive form is seldom used. The best way to evaluate a B-spline curve is to use the de Boor algorithm or the EvalBlossomProg algorithm.���2������(L�„M�.��� ,€ �€��0¤¨€�†"€���‚ÿ����1��›��RM�µO�–��� ú€9�€���¤€�‚‚€ �€�‚€�‚‚‚È)�ExecProgram("demo3.exe Full_Blossom", 0)�€�‡"€��à�‰€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ����Experiment with the B-Spline Curve����The following interactive demonstration uses the EvalBlossomProg routine to compute arbitrary blossom points, including (but not limited to) the B-spline curve.�������Click on the graphic to gain a deeper understanding of the behavior of B-spline curves.��Try setting the blossom values to �2.0�, �2.0�, �2.0� or �3.0�, �3.0�, �3.0,� and observe where the point lies.��*������„M�ßO�'��� €�€��0¤ª‚l€�‚ÿ���2������µO�€�.��� ,€ �€��0¤¨€ßO�€�CG��†"€���‚ÿ����*������ßO�G€�&��� €�€��0�¤¨€�‚‚ÿ����Ý��œ��€�$ƒ�A��� P€9�€���¤€ �€�‚€�‚€ �€�‚‚âá­õ…€�‰€�‚ÿ�Knot Insertion����A Bézier curve can be subdivided into two curves that together duplicate the original. This important operation is the basis for many routines for evaluating curves. The analog for B-splines is an operation called �knot insertion�. It amounts to creating a new knot sequence with an extra coincident knot that is equivalent to subdividing a B-spline curve into two.��From the point of view of the blossom form, this means generating a new set of blossom points with one new point and there discovering the blossom value associated with the new argument. This is the key to the method, since the new value follows from the ��blossoming principle.����ã��‘��G€�…�R��� r€%�€���¤€�‚€�€�€�€�€�€�‚‚‡"€��á�‚‚€�€�€�€�‚‚ÿ��Consider a simple example. Let �01�, �12�, and �23� be de Boor points in blossom form in the following figure:�����The valid parameter interval for the quadratic B-spline curve is [1, 2] (from �11� to �22�). Now seek a new knot (for example, at a parameter value of 1.5) that can be used to divide the curve into two segments. These two segments match the original curve. To accomplish this, find���H��� ���$ƒ�O…�(��� €@�€��˜¦‚l€�ƒ‚ÿ�1.�a new blossom sequence and��Š���`���…�Ù…�*��� $€À�€��¤‚l€�ƒ‚‚‚‚ÿ�2.�the corresponding blossom values.��The new sequence must be a shifted set of four values:������,���O…�Z†�U��� z€X�€��P�¤È€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ�{�0�,�1�; �1�,�1.5�; �1.5�,�2�; �2�,�3�}.��c��ï��Ù…�½ˆ�t��� ¶€ß�€���¤€�‚€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�€�‚ÿ��The blossom point �12� cannot be in this set, as it is not a shift of the knots. It is removed from the sequence. The new blossom values are easily derived from the now-familiar blossom principles. The blossom point �1�,�1.5� is on the line segment from �01� to �12�, an affine distance between �0� and �2�. Similarly �1.5�,�2� is on the line from �12� to �23�, an affine distance between �1� and �3�. This is illustrated in the following figure, which also shows the two new curve segments:��Õ��}��Z†�’Š�X��� ~€ý�€���¤€�‚‡"€��â�‚‚€�€�€�€�€�€�€�€�€�€�€�€�‚‚‚‚ÿ�����It should be clear that nonuniform B-splines are important in this process (if it isn't clear, look at the knot sequence for one of the curve segments: �0�,�1�; �1�,�1.5�; �1.5�,�2�). Even if the original points are uniform, the resulting knot insertion requires intermediate, unequally spaced values.��The two tasks generalize easily to any de Boor points in blossom form:���t���K���½ˆ�‹�)��� "€–�€��T˜î:‚l€�ƒ‚ÿ�1.�Create a new, sequentially shifted set of blossoms with the new knot.��o���G���’Š�u‹�(��� €Ž�€��Pì:‚l€�ƒ‚ÿ�2.�Find the positions of the blossoms with the blossoming principle.��-��ø��‹�¢�5��� 8€ó�€���¤€�‚‚‚‚‚‡"€��ã�‚‚‚ÿ��The first task admits only one obvious solution. It implies that some new knots are added (the same number as the degree of the curve), and some knots are removed (degree - 1 knots are removed).��Consider, for example, a nonuniform cubic B-spline:�����Inserting a knot at a parameter value of 2.4 divides the parameter range into two intervals. Putting the old sequence and the new sequence side by side often makes it clear which blossom values are deleted, which are new, and which remain the same:��'������u‹�É�$��� €�€���¤€�‚ÿ���Z������¢�#Ž�C���#V€.¥�8�±� ���€�€��˜€�‚ÿ�€€��˜‚ÿÿÿ�Old Knots��New Knots��^������É�Ž�N���#l€ ¥�8�±� ���€�€��˜ÿ��€€���€�‚ÿ�€€���‚ÿÿÿ��0,1,2��0,1,2��V������#Ž�׎�D���#X€$¥�8�±� ���€�€���€�‚ÿ�€€���€�‚ÿÿÿ�1,2,3���1,2,2.4��X������Ž�/�D���#X€(¥�8�±� ���€�€���€�‚ÿ�€€���€�‚ÿÿÿ�2,3,4.5���2,2.4,3��Z������׎�‰�D���#X€,¥�8�±� ���€�€���€�‚ÿ�€€���€�‚ÿÿÿ�3,4.5,6���2.4,3,4.5��M��� ���/�Ö�A���#R€¥�8�±� ���€�€���€�‚ÿ�€€���‚ÿÿÿ���3,4.5,6��Â���’���‰�¤À�0��� .€%�€���¤€�‚€�€�Ö�¤À�CG�€�‚‚ÿ���The red values clearly do not belong since they do not contain �2.4� and they do not shift correctly. The green values are the replacements.���2������Ö�ÖÀ�.��� ,€ �€��0¤¨€�†"€���‚ÿ������Z��¤À�cÂ�3��� 4€µ�€���¤€�‚‚€ �€�‚€�‚‚‚ÿ����What Has Been Accomplished in This Topic����The blossom principle has led naturally to B-splines, a type of curve used extensively in industry. This topic discussed the relationship between B-splines and Bézier curves, how the knot sequence modifies the curve, and how the characteristics of B-splines make them attractive as design tools.���2������ÖÀ�•Â�.��� ,€ �€��0¤¨€�†"€���‚ÿ����)������cÂ�¾Â�%��� €�€���¤€�‚‚ÿ����¼���r���•Â�zÃ�J��� d€ä�€��0¤¨‚l㤓h€�‰€�‚ã#jõ×€�‰€�‚ãS‹FZ€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: Rational Curves�����Go to the previous topic: Blossoms����<��� ���¾Â�¶Ã�1���ü��ÿÿÿÿÿÿÿÿN���¶Ã�ÿÿÿÿvÆ�EvalBlossom=������zÃ�óÃ�*��� $€&�€���˜˜B¤€ �€�‚ÿ�EvalBlossomProg���L���(���¶Ã�?Ä�$��� €P�€���¤€�‚ÿ�Recall the EvalBlossomProg procedure:��'������óÃ�fÄ�$��� €�€���œ€�‚ÿ���ª���w���?Ä�Å�3��� 6€î�€��P�œÈ€�€�€�€�€�‚‚‚ÿ�EvalBlossomProg  (�b�[i], . . ., �b�[i + d], u[1], . . ., u[d], t[i], . . ., t[i + 2d - 1])��for k = 0 to (d - 1) do��D������fÄ�TÅ�&��� €<�€��P�œ‘€€�‚ÿ�for j = 0 to (d - k - 1) do��¶���|���Å� Æ�:��� D€ø�€��P�œÙ€€�‚‚€�€�€�€�€�€�‚ÿ�Beta = (u[k + 1] - t[i + k + j]) / (t[i + d + j] - t[i + k + j])�Alpha = 1 - Beta��b�[j] = Alpha �b�[j] + Beta �b�[j + 1]��,������TÅ�6Æ�&��� € �€��P�œ‘€€�‚ÿ�end��@������ Æ�vÆ�,��� (€(�€��P�œÈ€�‚€�€�‚ÿ�end�return �b�[0]��8������6Æ�®Æ�1���/��ÿÿÿÿÿÿÿÿO���®Æ�ÿÿÿÿ¥Ç�de Boor6������vÆ�äÆ�'��� €�€���˜˜B˜€ �‚ÿ�Carl de Boor��Á���—���®Æ�¥Ç�*��� "€/�€���œ€�€�‚ÿ�Carl de Boor, an American mathematician, developed the�special case of EvalBlossomProg when the u(i) parameters�are equal. It evaluates a B-spline.���C������äÆ�èÇ�1���Z��ÿÿÿÿÿÿÿÿP���èÇ�ÿÿÿÿÿÈ�The B-Spline Curve<������¥Ç�$È�'��� €*�€���˜˜B˜€ �‚ÿ�The B-Spline Curve��Û���±���èÇ�ÿÈ�*��� "€c�€���œ€�€�‚ÿ�A spline was originally a strip made of wood or metal that�is used to create a smooth curve through a set of points.�This was the original method for creating smooth curves.���>��� ���$È�=É�1���û���ÿÿÿÿÿÿÿÿQ���=É�ÿÿÿÿúÉ�C2 Continuity7������ÿÈ�tÉ�'��� € �€���˜˜B˜€ �‚ÿ�C2 Continuity��†���^���=É�úÉ�(��� €¼�€���œ€�€�‚ÿ�Recall C2 continuity: This means that the second�derivatives of the curves are continuous.���<��� ���tÉ�6Ê�1�����ÿÿÿÿÿÿÿÿR���6Ê�ÿÿÿÿË�Subdivision9������úÉ�oÊ�*��� $€�€���˜˜B˜€ �€�‚ÿ�Subdivision��� ���x���6Ê�Ë�(��� €ð�€���œ€�€�‚ÿ�Subdivision is the process, usually through reparameterization,�of separating a single curve segment into two parts.���6������oÊ�EË�1���Ü��ÿÿÿÿÿÿÿÿS���EË�ÿÿÿÿëÌ�Knots/������Ë�tË�'��� €�€���˜˜B˜€ �‚ÿ�Knots��w��J��EË�ëÌ�-��� (€•�€���œ€�€�‚ÿ�EvalBlossomProg has introduced new quantities called�knots. Since the B-spline uses fewer control points to�define curve segments, there are clearly some degrees�of freedom available for definition. These are the knots.�Their geometric significance will become more apparent�after the discussion on nonuniform B-spline curves.���7������tË�"Í�1���Æ���ÿÿÿÿÿÿÿÿT���"Í�ÿÿÿÿ±Í�Answer0��� ���ëÌ�RÍ�'��� €�€���˜˜B˜€ �‚ÿ�Answer��_���-���"Í�±Í�2��� 4€\�€���œ€�‚‚‡"€��ä�€�‚ÿ�The following figure gives the answer:������7������RÍ�èÍ�1���/��ÿÿÿÿÿÿÿÿU���èÍ�ÿÿÿÿàÎ�Answer0��� ���±Í�Î�'��� €�€���˜˜B˜€ �‚ÿ�Answer��È���ˆ���èÍ�àÎ�@��� N€�€���œ€�€�€�€�€�‚‚‡"€��å�€�‚ÿ�The following figure gives the answer. The points �000� and �111��define a portion of the curve that is not part of the B-spline.������B������Î�"Ï�1���÷���ÿÿÿÿÿÿÿÿV���"Ï�ÿÿÿÿ � �Nonuniform Curves;������àÎ�]Ï�'��� €(�€���˜˜B˜€ �‚ÿ�Nonuniform Curves��z���R���"Ï� � �(��� €¤�€���œ€�€�‚ÿ�Nonuniform curves have a knot sequence�in which the knots are unevenly spaced.��������������������������������������������]Ï� � �àÎ�I������]Ï�U� �1���ô��ÿÿÿÿÿÿÿÿW���U� �ÿÿÿÿ� �The Blossoming PrincipleF������ � �›� �*��� $€8�€���˜˜B˜€ �€�‚ÿ�The Blossoming Principle���/��ò���U� �Ê �=��� H€å�€���¤€�‚‚€�€�€�€�€�€�‚‚ÿ�The digit that differs between two blossoms is replaced�with another value, giving the affine distance along the�line segment for the new value.��Thus �axx� is found on the line segment between �bxx� and��cxx�, an affine distance given by���6������›� �� �1��� 2€ �€��P�¤È€�†"€��æ�€�‚ÿ�����<��� ���Ê �< �1��� ��ÿÿÿÿÿÿÿÿX���< �ÿÿÿÿ  �ConvolutionB������� �~ �*��� $€0�€���˜˜B˜€ �€�‚ÿ�Integral Convolution���O���*���< �Í �%��� €T�€���œ€�‚‚ÿ�The convolution of f and g is given by���<������~ �  �7��� >€�€��P�œÐ€�†"€��ç�†"€��è�‚ÿ�����@������Í �I �1���~0��üƒ�q�Y���I �‹ �b �Rational CurvesB������  �‹ �'��� €6�€���˜˜B¤€ �‚ÿ�Topic 7: Rational Curves��F��� ���I �Ñ �&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn��•���a���‹ �f �4��� 8€Â�€��2˜¦¨‚l€ �€�ƒ‚€ �€�ƒ‚ÿ�·��why a weight can be associated with a control point,��·��the advantages of rational curves.��'������Ñ � �$��� €�€���¤€�‚ÿ���1������f �¾ �-��� *€ �€��¤€�†"€���‚ÿ������Î�� �À �4��� 6€�€���¤€�‚‚€ �€�‚€�‚‚‚‚ÿ����Introduction����A polynomial curve can exist in any dimension; it depends only on the number of dimensions used to describe the control points. Usually curves in CAGD are of two or three dimensions. The class of curves studied so far can be extended by considering projections of curves of a given dimension to a dimension lower by one. For example, a three-dimensional curve can be projected onto a two-dimensional plane.��The perspective projection is one familiar to programmers of computer graphics. Points in n-space are taken to a plane parallel to n - 1 axes, but displaced by a unit interval. A point is then projected along a straight line through the origin to the point of intersection with the plane.��,��ö���¾ �ì �6��� :€ï�€���¤€�‚‚‚‡"€��é�‚‚‚‚‚‚ÿ��The following figure illustrates the projection from n = 3 to n = 2:�����The idea generalizes for any n. For purposes of clarity, the projection from three dimensions to two dimensions will be used in this discussion.��Any point on the curve���2������À � �.��� ,€ �€��P�¤È€�†"€��ê�‚ÿ����D������ì �b �&��� €<�€���¤€�‚‚‚ÿ��will project to the point���=��� ��� �Ÿ �3��� 6€�€��P¤È‚¡…€�†"€��ë�ƒ‚ÿ���(7.1)��<����b �Û �4��� 6€�€���¤€�‚‚‚âã…§/€�‰€�‚ÿ��Points on the x, y plane where x and y are not zero are said to project to infinity.��The new, two-dimensional curve now has two coordinates found by dividing the x and y coordinates by the z coordinate. It is called a ��rational curve.�� The set of rational curves includes all polynomial curves. This is seen by letting z(t) = 1. The set also includes all conic sections, that is, ellipses, parabolas, and hyperbolas. Except for the parabola, conic sections cannot be written as polynomial curves. In many applications exact conic sections are required. For instance, think of how often a circular arc is used in engineering. The support of conic sections was the main motive driving the development of rational curves. The rational curve also offers design flexibility.��'������Ÿ � �$��� €�€���¤€�‚ÿ���1������Û �3 �-��� *€ �€��¤€�†"€���‚ÿ����ó���¾��� �& �5��� 8€}�€���¤€�‚‚€ �€�‚€�‚‚‚‚‚ÿ����The Two-Dimensional Rational Bézier Curve����A special notation is employed for writing the control points of a rational Bézier curve.��Start with a curve in three dimensions given by���2������3 �X �.��� ,€ �€��P�¤È€�†"€��ì�‚ÿ����Ú���±���& �2 �)��� €c�€���¤€�‚‚‚‚‚ÿ��Any control point in 3-space can be written by simply choosing the values of x and y appropriately; w is the z-coordinate.��From the projection above, the rational curve is���=��� ���X �o �3��� 6€�€��P¤È‚¡…€�†"€��í�ƒ‚ÿ���(7.2)��l���D���2 �Û �(��� €ˆ�€���¤€�‚‚‚‚‚ÿ��The numerator and denominator always match in degree.��Now, let���2������o �@ �.��� ,€ �€��P�¤È€�†"Û �@ �  �€��î�‚ÿ����z���T���Û �“@ �&��� €¨�€���¤€�‚‚‚ÿ��Now the rational Bézier curve may be described with more conventional notation:���=��� ���@ �Ð@ �3��� 6€�€��P¤È‚¡…€�†"€��ï�ƒ‚ÿ���(7.3)��°��y��“@ �€B �7��� <€ó�€���¤€�‚â˄ѱ€�‰€�‚‚‚‚‚‚ÿ��To define a rational Bézier curve, give a control polygon as before, and with each control point, associate a value w, called its ��weight.����The next interactive demonstration shows a quadratic rational Bézier curve projection and its 2-space equivalent.��While observing this demonstration, notice what happens when the weight values are changed. In particular, notice:���Z���.���Ð@ �ÚB �,��� (€\�€��2˜¦¨‚l€ �€�ƒ‚ÿ�·��What happens when all the weights are 1?��?������€B �C �-��� *€$�€��6˜˜¦¨‚l€ �€�ƒ‚ÿ�·��Try setting:��3������ÚB �LC �/��� .€ �€��P�¤‘€€�†"€��ð�‚ÿ����?������C �‹C �-��� *€$�€��6˜˜¦¨‚l€ �€�ƒ‚ÿ�·��Try setting:��3������LC �¾C �/��� .€ �€��P�¤‘€€�†"€��ñ�‚ÿ����D��×���‹C �E �m��� ¨€±�€���¤€�‚È1�ExecProgram("demo4.exe Quad_Rational_Bezier", 0)�€�‡"€��ò�‰€�‚‚‚‚ÿ�������Click on the graphic to run the quadratic rational Bézier interactive demonstration.��The third suggestion above indicates that there is some redundancy in the choice of weights. The same curve is given by���2������¾C �4E �.��� ,€ �€��P�¤È€�†"€��ó�‚ÿ����0��� ���E �dE �&��� €�€���¤€�‚‚‚ÿ��as by���2������4E �–E �.��� ,€ �€��P�¤È€�†"€��ô�‚ÿ����½���–���dE �SF �'��� €-�€���¤€�‚‚‚ÿ��The curve in 3-space simply scales without changing the projection. The redundancy can be eliminated by standardizing the quadratic curve so that���2������–E �…F �.��� ,€ �€��P�¤È€�†"€��õ�‚ÿ����Œ���[���SF �G �1��� 2€¸�€���¤€�‚‚‚†"€��ö�‚‚ÿ��Given any set of weights, the formula for standardizing the curve is the following:������2������…F �CG �.��� ,€ �€��P�¤È€�†"€��÷�‚ÿ����.������G �qG �&��� €�€���¤€�‚‚‚ÿ��and���2������CG �£G �.��� ,€ �€��P�¤È€�†"€��ø�‚ÿ����®������qG �QH �-��� (€�€���¤€�‚€�€�‚‚ÿ��This replacement does not change the form of the curve. Finally, by dividing through by w�2�, the standard form is obtained.���1������£G �‚H �-��� *€ �€��¤€�†"€���‚ÿ����¼���‰���QH �>I �3��� 4€�€���¤€�‚‚€ �€�‚€�‚‚‚ÿ����Finding the Conic Sections����From a rational quadratic Bézier curve in standard form, it is easy to obtain the conic sections. If���2������‚H �pI �.��� ,€ �€��P�¤È€�†"€��ù�‚ÿ����F��� ���>I �¶I �&��� €@�€���¤€�‚‚‚ÿ��then a parabola results. If���2������pI �èI �.��� ,€ �€��P�¤È€�†"€��ú�‚ÿ����K���%���¶I �3J �&��� €J�€���¤€�‚‚‚ÿ��then the curve is an ellipse. If���2������èI �eJ �.��� ,€ �€��P�¤È€�†"€��û�‚ÿ����š���r���3J �ÿJ �(��� €ä�€���¤€�‚‚‚‚‚ÿ��then the curve is a hyperbola.��The case of the circle deserves special attention. Use the ellipse condition,���2������eJ �1K �.��� ,€ �€��P�¤È€�†"€��ü�‚ÿ����ˆ���Q���ÿJ �¹K �7��� >€¤�€���¤€�‚€�€�‚‚†"€��ý�‚‚ÿ��but the control points �b� also play a role. The condition is as follows.������2������1K �ëK �.��� ,€ �€��P�¤È€�†"€��þ�‚ÿ����Ô���­���¹K �¿L �'��� €[�€���¤€�‚‚‚ÿ��Because of the symmetry of the circle, the control points must also be configured symmetrically as an isosceles triangle. The following interactive demonstration forces���2������ëK �ñL �.��� ,€ �€��P�¤È€�†"€��ÿ�‚ÿ����‰��(��¿L �zN �a��� €S�€���¤€�‚‚‚È#�ExecProgram("demo4.exe Conics", 0)�€�‡"€���‰€�‚‚ÿ��The value of the intermediate weight may be varied to obtain a parabola, an ellipse, and a hyperbola.�������Click on the graphic to study the way that the quadratic rational Bézier curve produces conics.��Use the "Ellipse" button to set the weights and control points for the conic sections.���1������ñL �«N �-��� *€ �€��¤€�†"€���‚ÿ����Z��Û��zN � ���� Ì€¹�€���¤€�‚‚€ �€�‚€�‚€�€�€ �€�‚€�‚È,�ExecProgram("demo4.exe Rational_Bezier", 0)�‡"€��‰€�‚ÿ����The General Rational Bézier Curve����As alluded to previously, the control points �b� may be of any dimension. Add one more coordinate, the weight, to define the curve in a higher dimensional space. This space «N � �  �is called the �projective space�. To obtain the rational curve, project the polynomial curve onto the hyperplane. Behavior and properties of the rational curve derive from this operation.�������Click on the graphic to investigate cubic rational Bézier curves.��¡���{���«N �² �&��� €ö�€���¤€�‚‚‚ÿ��The first thing that becomes apparent when studying the general case of the rational Bézier curve is that the equation���2������ �ä �.��� ,€ �€��P�¤È€�†"€��‚ÿ����>������² �"‚ �&��� €0�€���¤€�‚‚‚ÿ��can be rewritten as���=��� ���ä �_‚ �3��� 6€�€��P¤È‚¡…€�†"€��ƒ‚ÿ���(7.4)��0��� ���"‚ �‚ �&��� €�€���¤€�‚‚‚ÿ��where���=��� ���_‚ �Ì‚ �3��� 6€�€��P¤È‚¡…€�†"€��ƒ‚ÿ���(7.5)�� ��ç���‚ �Ùƒ �&��� €Ï�€���¤€�‚‚ÿ��Investigation of this newly defined set of basis functions reveals partition of unity, convex hull, and endpoint interpolation properties. The rational Bézier curve also has all the properties from the polynomial form, that is,��µ���q���Ì‚ �Ž„ �D��� X€â�€��2˜¦¨‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��affine invariance,��·��tangent interpolation at endpoints,��·��variation diminishing,��·��linear precision.����î��Ùƒ �¤† �(��� €Ý�€���¤€�‚‚‚‚ÿ��The important de Casteljau algorithm can be applied to the rational curve by applying it in projective space. That is, the curve is evaluated in one dimension higher by de Casteljau and then the projection is performed. This involves the division by the weight coordinate. This procedure may also be used to subdivide the curve.��The only reservation to this approach is that the final divide may cause numerical instability. [Farin93] gives an alternative with greater numerical stability.��¹��y��Ž„ �]‰ �@��� N€÷�€���¤€�‚‚‚†"€��‚‚‚‚†"€��‚‚‚ÿ��The weights of a rational curve can add some design flexibility. The movement of a control point moves any point on the curve in a parallel direction, as shown in the following figure:�����However, if only the weight is changed, not the control point, then the points on the curve move along a line toward the associated control point:�����Besides the ability to handle conics and to gain design flexibility, the rational form allows freedom through the weights to solve certain other problems. Such problems include higher order interpolatory reparameterization without changing the curve shape, and curve fairing, or smoothing.��ö���Œ���¤† �SŠ �j��� ¢€�€���¤€�‚È0�ExecProgram("demo4.exe Pts_Rational_Bezier", 0)�€�‡"€��‰€�‚‚ÿ�������Click on the graphic to observe how points on the Bézier curve move when the control points are moved or the weights are adjusted.���1������]‰ �„Š �-��� *€ �€��¤€�†"€���‚ÿ����Ì��™��SŠ �PŒ �3��� 4€3�€���¤€�‚‚€ �€�‚€�‚‚‚ÿ����Rational B-Spline Curves����The rational B-spline curve is defined in direct analogy to the rational Bézier; it is a projection of the B-spline curve to a space of lower dimension. This makes obvious sense given that the B-spline curve can be converted to Bézier form. If this is done in projective space, then the rational B-spline curve and the rational Bézier curve can be converted into each other:���2������„Š �‚Œ �.��� ,€ �€��P�¤È€�†"€��‚ÿ����S���-���PŒ �ÕŒ �&��� €Z�€���¤€�‚‚‚ÿ��The form of a rational B-spline curve is���=��� ���‚Œ � �3��� 6€�€��P¤È‚¡…€�†"€�� ƒ‚ÿ���(7.6)��g��7��ÕŒ �y �0��� .€o�€���¤€�‚€�€�‚‚‚‚‚ÿ��Here, the �d� are the de Boor points and N(u), the B-spline basis functions. L is the number of segments. As with the Bézier curve, the w values represent the weights associated with the de Boor points.��The most general form of a curve discussed thus far is the rational form of the nonuniform B-spline. It is called a NURBS (non-uniform rational B-spline). It is described by control (de Boor) points, a set of weights, and a knot sequence.��Operations and properties follow for the NURBS from the B-spline in the same way as seen for the rational Bézier curve.��'������ �  �$��� €�€���¤€�‚ÿ���1������y �Ñ �-��� *€ �€��¤€�†"€���‚ÿ����H����  �%Á �2��� 2€-�€���¤€�‚€ �€�‚€�‚Ñ �%Á �  �‚‚ÿ���What Has Been Accomplished in This Topic����Rational curves extend the capabilities of standard Bézier and B-spline curves, and permit conic sections to be realized. With the introduction of further degrees of freedom, the designer has a tool to solve additional problems.���1������Ñ �VÁ �-��� *€ �€��¤€�†"€���‚ÿ����)������%Á �Á �%��� €�€���¤€�‚‚ÿ����¼���r���VÁ �; �J��� d€ä�€��0¤¨‚l㤓h€�‰€�‚ã*/Aµ€�‰€�‚ã!}¤÷€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: Surfaces�����Go to the previous topic: B-Spline Curves����'������Á �b �$��� €�€���¤€�‚ÿ���@������; �¢Â �1�����ÿÿÿÿÿÿÿÿZ���¢Â �ÿÿÿÿvà �Rational Curves=������b �ß �*��� $€&�€���˜˜B˜€ �€�‚ÿ�Rational Curves���—���o���¢Â �và �(��� €Þ�€���¦€�€�‚ÿ�Rational curves are so named because they are written�as a ratio. They are the quotient of two polynomials.���7������ß �­Ã �1���ø���ÿÿÿÿÿÿÿÿ[���­Ã �ÿÿÿÿnÄ �Weight0��� ���và �Ýà �'��� €�€���˜˜B˜€ �‚ÿ�Weight��‘���h���­Ã �nÄ �)��� "€Ð�€���¦€�€�‚ÿ�The weight w of a control point is a�measure of the contribution the point�makes to the final curve.���9������Ýà �§Ä �1���f*��]�‹�\���§Ä �âÄ �O �Surfaces;������nÄ �âÄ �'��� €(�€���˜˜B˜€ �‚ÿ�Topic 8: Surfaces��F��� ���§Ä �(Å �&��� €@�€��2�˜œª€�‚ÿ�In this topic, you will learn��³���v���âÄ �ÛÅ �=��� J€ì�€��r˜ìª:‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��the characteristics of surfaces,��·��the use of Bézier surface patches,��·��the use of B-spline surface patches.��'������(Å �Æ �$��� €�€���¤€�‚ÿ���1������ÛÅ �3Æ �-��� *€ �€��¤€�†"€���‚ÿ����)������Æ �\Æ �%��� €�€���¤€�‚‚ÿ����K����3Æ �§È �C��� T€�€���œ€ �‚€�‚‚‚†"€�� ‚‚â#ÑÈ’€�‰€�‚ÿ�Introduction to Surfaces���Imagine moving the set of control points of the Bézier curve in three dimensions. As they move in space, new curves are generated. If they are moved smoothly, then the curves formed create a surface, which may be thought of as a bundle of curves. If each of the control points is moved along a Bézier curve of its own, then a Bézier surface patch is created.�����This can be described by changing the control points in the ��Bézier formula�� into Bézier curves; thus a surface is defined by��'������\Æ �ÎÈ �$��� €�€���œ€�‚ÿ���?��� ���§È � É �5��� :€�€��pœß€È‚¡…€�†"€�� ƒ‚ÿ���(8.1)��{��K��ÎÈ �ˆÊ �0��� .€—�€���œ€�‚€�€�€�‚‚ÿ��Notice there is one parameter for the control curves and one for the "swept" curve. It is convenient to write the control curves as Bézier curves that have the same degree as the original control curves. Given that the ith control point has control points �b�ij�, then the surface given in equation 8.1 above can be written as���?��� ��� É �ÇÊ �5��� :€�€��pœß€È‚¡…€�†"€�� ƒ‚ÿ���(8.2)��8�� ��ˆÊ �ÿË �+��� $€�€���œ€�‚‚‚‚‚‚‚ÿ��where m is the degree of the control curves.��Such a surface can be thought of as nesting one set of curves inside another. From this simple characteristic, many properties and operations for surfaces may be derived.��Simple algebra changes equation 8.2 above into���?��� ���ÇÊ �>Ì �5��� :€�€��pœß€È‚¡…€�†"€�� ƒ‚ÿ���(8.3)��³���Œ���ÿË �ñÌ �'��� €�€���œ€�‚‚‚ÿ��That is, even though one curve was swept along the other, there is no preferred direction. The surface patch could have been written as���=��� ���>Ì �.Í �3��� 6€�€��PœÈ‚¡…€�†"€��ƒ‚ÿ���(8.4)��0��� ���ñÌ �^Í �&��� €�€���œ€�‚‚‚ÿ��where���=��� ���.Í �›Í �3��� 6€�€��PœÈ‚¡…€�†"€��ƒ‚ÿ���(8.5)��½��‡��^Í �XÏ �6��� :€�€���œ€�‚‚‚‚‚†"€��‚‚‚‚ÿ��The curve is simply swept in the other direction.��The set of control points forms a rectangular control mesh. A 3 by 3 (bicubic) control mesh is shown here:�����There are 16 control points in the bicubic control mesh. In general there will be (n + 1) by (m + 1) control points. By convention, the i index is associated with the u parameter, and the j index with the v parameter. Hence,���2������›Í �ŠÏ �.��� ,€ �€��P�œÚ€�†"€��‚ÿ����A������XÏ �ËÏ �&��� €6�€���œ€�‚‚‚ÿ��gives the Bézier curve���=��� ���ŠÏ �� �3��� 6€�€��PœÈ‚¡…€�†"€��ƒ‚ÿ��ËÏ �� �nÄ ��(8.6)�� ��×���ËÏ � �2��� 2€±�€���œ€�‚†"€��‚‚‚‚ÿ�����Each marginal set of control points defines a Bézier curve (the four border curves), and each of these curves is a boundary of the Bézier surface patch. Such a curve is shown in blue in the preceding figure.���1������� �N �-��� *€ �€��¤€�†"€���‚ÿ�������Ð��� �N �0��� .€¡�€���œ€�‚‚€ �‚€�‚‚‚ÿ����Properties of the Bézier Surface Patch���Many of the properties of the Bézier surface are derived directly from those of the Bézier curve, especially those curves that form the boundaries of the patch.���H������N �– �,��� (€8�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Endpoint Interpolation�� ���n���N �6 �2��� 4€Þ�€��ì‚´€�ƒ†"€��‚ÿ�The Bézier surface patch passes through all four corner control points. Formally, for the bicubic case,������'������– �] �$��� €�€���œ€�‚ÿ���D������6 �¡ �,��� (€0�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Tangent Conditions��1�� ��] �Ò �%��� €�€���ì€�‚ÿ�The four border curves of the Bézier surface patch are cotangent to the first and last segments of each border control polygon, at the first and last control points. The normal to the surface patch at each vertex may be found from the cross product of the tangents.��'������¡ �ù �$��� €�€���œ€�‚ÿ���=������Ò �6 �,��� (€"�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Convex Hull��°���t���ù �æ �<��� H€è�€���ì€�€ �€�€ �€�€ �€�€ �€�‚ÿ�The Bézier surface patch is contained in the convex hull of its control mesh for 0 �£� u �£� 1 and 0 �£� v �£� 1.��'������6 �  �$��� €�€���œ€�‚ÿ���C������æ �P �,��� (€.�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Affine Invariance����ú���  �o �%��� €õ�€���ì€�‚ÿ�The Bézier surface patch is affinely invariant with respect to its control mesh. This means that any linear transformation or translation of the control mesh defines a new patch that is just the transformation or translation of the original patch.��'������P �– �$��� €�€���œ€�‚ÿ���G������o �Ý �,��� (€6�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Variation Diminishing��‹���g���– �h �$��� €Î�€���ì€�‚ÿ�Although this is difficult to define for surfaces, the control mesh suggests the shape of the patch.��'������Ý � �$��� €�€���œ€�‚ÿ���B������h �Ñ �,��� (€,�€��T˜î:‚l€ �€�ƒ‚ÿ�·��Planar Precision����ø��� �î �%��� €ñ�€���ì€�‚ÿ�The Bézier surface patch has planar precision: if all the points in the control mesh lie in a plane, the surface patch will lie in the same plane; if all the points in the control mesh form a straight line, the surface is also reduced to a line.��5��î��Ñ �# �G��� \€Ý�€���œ€�‚‚€ �‚€�‚â‹¥À€�‰€�â¥Ô©€�‰€�‚ÿ����Evaluation of the Bézier Surface Patch���As with the properties described above, the evaluation of a Bézier surface patch can also be derived from the Bézier curve. To evaluate a point on the patch at parameter value (u, v), apply the ��de Casteljau algorithm�� in a nested fashion similar to ��equation 8.3.�� That is, first evaluate the control curves in the u direction, which reduce to control points in the v direction. These points are again evaluated with de Casteljau's algorithm.����Þ��î �< �;��� D€¿�€���œ€�‚†"€��‚‚‚‚‚‚‚‚‚€ �‚ÿ�����In the preceding figure, a point on a biquadratic Bézier surface patch is evaluated by first computing three points in u, then one point in v. There are nine points (3 by 3) in the control mesh.��As explained earlier, the order of u and v is not important: the same point is generated by evaluating in v first, and then in u.��Any other evaluation technique used for Bézier curves may be applied in a nested fashion to surfaces.����Subdivision of the Bézier Surface Patch����Ç��# �N@ �?��� L€‘�€���œ€�‚‚‚†"€��‚‚â]˵"€�‰€�‚ÿ��As with the Bézier curve, the de Casteljau algorithm can be applied to subdivide a Bézier surface patch. When a surface patch is subdivided, it yields four subpatches that share a corner at the (u, v) subdivision point.�����Recall that when a curve was subdivided, the new curve's control points appeared as the legs of a ��systolic array.�� In the surface case, subdividing each row of < �N@ �nÄ �the control mesh produces points of the systolic array for each.��µ��€��< �C �5��� 8€�€���œ€�‚‚‚†"€��‚‚‚‚‚ÿ��Each point on each leg of every row's systolic array now becomes a control point for a columnar set. A biquadratic case is shown here:�����In this case, three points in each row produce five points after subdivision.��Now consider the points in columns, subdividing the columns with de Casteljau's algorithm. The points in the legs of their systolic arrays become the control points of the new subpatches. In the preceding example, rows with three control points produce five "leg" points, that is, five columns of three points. Each column then produces five control points; so, a 3 by 3 grid generates a 5 by 5 grid after subdivision:��ü��®��N@ �ÿD �N��� j€c�€���œ€�‚†"€��‚‚‚‚†"€��‚‚‚‚†"€��‚‚‚‚‚‚ÿ�����The control meshes of the four new patches are produced as follows:�����The central row and column of control points are shared by each 3 by 3 subpatch:�����The order of the scheme does not matter. Columns may have been taken first, and then rows.��Subdivision is a basic operation of surfaces. Many "divide and conquer" algorithms are based on it. Imagine clipping a surface to a viewing window with the two properties of���^���*���C �]E �4��� 8€T�€��T˜î:‚l€ �€�ƒ‚€ �€�ƒ‚ÿ�·��convex hull property��·��subdivision��¬���z���ÿD � F �2��� 4€ô�€���œ€�‚âýžè€�‰€�‚‚ÿ��Hint: It is much easier to test the convex hull of a patch against a viewing window than the patch itself. ��Answer�����1������]E �:F �-��� *€ �€��¤€�†"€���‚ÿ����º���Š��� F �ôF �0��� .€�€���œ€�‚‚€ �‚€�‚‚‚ÿ����Uniform B-Spline Surfaces���As with the Bézier surface, the B-spline surface is defined as a nested bundle of curves, thus yielding���=��� ���:F �1G �3��� 6€�€��PœÈ‚¡…€�†"€��ƒ‚ÿ���(8.7)��0��� ���ôF �aG �&��� €�€���œ€�‚‚‚ÿ��where���3������1G �”G �/��� .€ �€��¤‚l€�†"€��‚ÿ����¡���n���aG �5H �3��� 6€Ü�€��˜¤‚l€ �€�ƒ‚€ �€�ƒ‚ÿ�·��n, m are the degrees of the B-splines,��·��L, M are the number of segments, so there are L by M patches.��f��=��”G �›I �)��� €{�€���œ€�‚‚‚‚‚ÿ��All operations used for B-spline curves carry over to the surface via the nesting scheme, including knot insertion, de Boor's algorithm, and so on.��B-spline curves are especially convenient for obtaining continuity between polynomial segments. This convenience is even stronger in the case of B-spline surfaces:���‹���R���5H �&J �9��� B€¦�€��T˜î8‚l€�†"€��‚€ �€�ƒ‚ÿ����·��B-splines define quilts of patches with corresponding design flexibility.��ï��¹��›I �L �6��� :€u�€���œ€�‚‚‚‚‚‚‚†"€��‚‚ÿ��B-spline curves are more compact at representing a design than Bézier curves. This advantage is "squared" in the case of surfaces.��These advantages are tempered by the fact that operations are typically more efficient on Bézier curves. Conventional wisdom says that it is best to design and represent surfaces as B-splines, and then convert to Bézier form for operations.��The following figure shows a simple B-spline surface patch:������1������&J �FL �-��� *€ �€��¤€�†"€���‚ÿ����¦��t��L �ìM �2��� 2€é�€���œ€�‚‚€ �‚€�‚‚‚‚‚ÿ����What Has Been Accomplished in This Topic���The majority of industrial design tasks require the creation of three-dimensional products. Bézier and B-spline surface patches supply the tools for such products.��This topic has introduced descriptions for these surface patches and given their advantages and behavior. Subdivision of the Bézier patch was also presented.���1������FL �N �-��� *€ �€��¤€�†"€���‚ÿ����)������ìM �FN �%��� €�€���œ€�‚‚ÿ����Í������N �O �L��� f€�€��pì¨:‚l㤓h€�‰€�‚ãôì%€�‰€�‚ã#jõ×€�‰€�‚ÿ��Go to the Table of Contents�����Go to the next topic: Images and Applications�����Go to the previous Topic: Rational Curves����C������FN �VO �1���Ñ��ÿÿÿÿÿÿÿÿ]���VO �ÿÿÿÿ‚ �Clipping a Surface<������O �’O �'��� €*�€���˜˜B˜€ �‚ÿ�Clipping a Surface��K���&���VO �ÝO �%��� €L�€���œ€�‚‚ÿ�Recursively perform the following:���-��Ý��’O �‚ �P��� n€»�€��T˜î:‚ÝO �‚ �O �l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��Test the convex hull of the patch against the viewing window;�if clipping is necessary, then continue, else exit.��·��Subdivide the patch at u = 0.5, v = 0.5; test the convex hulls�of each subpatch against the viewing window.��·��Render those patches that intersect the viewing window;�ignore those patches outside the viewing window.��·��Recurse, subdividing patches that still intersect the window.��·��Continue until the patches are beneath a certain preset threshold.��U���$���ÝO �k‚ �1���Ú���ÿÿÿÿÿÿÿÿ^���k‚ �ÿÿÿÿð‚ �Recall de Casteljau's Systolic ArrayR���(���‚ �½‚ �*��� $€P�€���˜˜B˜€ �€�‚ÿ�Recall de Casteljau's Systolic Array���3������k‚ �ð‚ �.��� ,€ �€���¦€�‚†"€��m�‚ÿ�����S���"���½‚ �Cƒ �1���†��ÿÿÿÿÿÿÿÿ_���Cƒ �ÿÿÿÿv„ �Recall de Casteljau's Algorithm...M���#���ð‚ �ƒ �*��� $€F�€���˜˜B˜€ �€�‚ÿ�Recall de Casteljau's Algorithm���æ���¿���Cƒ �v„ �'��� €�€���¦€�‚ÿ�De Casteljau's algorithm evaluates points on a Bézier curve by�a process of recursive linear interpolation. At each successive�level in the iteration, the algorithm converges to the curve.��J������ƒ �À„ �1���ü���ÿÿÿÿÿÿÿÿ`���À„ �ÿÿÿÿr… �The Bezier Surface Patch:U���+���v„ �… �*��� $€V�€���˜˜B˜€ �€�‚ÿ�The Bézier Surface Patch, Equation 8.3:���'������À„ �<… �$��� €�€���¦€�‚ÿ���6������… �r… �1��� 2€ �€��P�¦È€�†"€��€�‚ÿ�����?������<… �±… �1���ÿ���ÿÿÿÿÿÿÿÿa���±… �ÿÿÿÿq† �Bezier Formula8������r… �é… �'��� €"�€���˜˜B˜€ �‚ÿ�Bézier Formula��V���1���±… �?† �%��� €b�€���¦€�‚‚ÿ�The equation for the standard Bézier curve is���2������é… �q† �.��� ,€ �€��P�¦È€�†"€�� ‚ÿ����@������?† �±† �1���R&��q�ò�b���±† �û† �@ �Other QuestionsJ���#���q† �û† �'��� €F�€���˜˜B˜€ �‚ÿ�Topic 9: Images and Applications��F��� ���±† �A‡ �&��� €@�€��2�˜¤ª€�‚ÿ�In this topic, you will learn��ì���¯���û† �-ˆ �=��� H€_�€��2˜¤ª‚l€ �€�ƒ‚€ �€�ƒ‚€ �€�ƒ‚ÿ�·��tools to evaluate the properties of a surface,��·��how CAGD assists design of industrial products,��·��uses of CAGD that are creative and artistic as well as functional.��*������A‡ �Wˆ �'��� €�€��0¤ª‚l€�‚ÿ���2������-ˆ �‰ˆ �.��� ,€ �€��0¤¨€�†"€���‚ÿ����,������Wˆ �µˆ �(��� €�€��0¤ª‚l€�‚‚ÿ����7������‰ˆ �ìˆ �'��� € �€���¤€ �€�‚ÿ�Introduction���ƒ��Y��µˆ �oŠ �*��� "€³�€��0¤ª‚l€�‚‚‚ÿ��The best method of evaluating the properties of an object's surface, apart from actually manufacturing the object, is to visualize the object with a computer. This method is commonly applied for engineering, analysis, styling, marketing, and in the conceptual stages of a project. This topic presents examples of some of these applications.���2������ìˆ �¡Š �.��� ,€ �€��0¤¨€�†"€���‚ÿ����·��q��oŠ �X �F��� Z€å�€��0¤ª‚l€�‚‚€ �‚€�‚‚‚‚‡"€��!€�€�‚ÿ����Evaluating Surface Characteristics: Reflection Maps���Imagine a bank of parallel fluorescent tubes illuminating an object. The reflections of the lights flow across the surface and accurately indicate the continuity between surface patches. The following three figures demonstrate this method using two surface patches of different continuity across the boundary.����G0 Continuity����Here, the patches are connected with positional (G0) continuity. The reflection lines indicate discontinuous jumps. The reflections trace the behavior of the surface normals, which are discontinuous at the join between the two patches.��,��Ù��¡Š �„ �S��� t€·�€��0¤ª‚l€�‚‚‚‚‡"€��"€�€�‚‚‚‚‚‡"€��#€�€�‚ÿ�Courtesy of Parametric Technology Corp. © 1995�Data generated by Pro/CDRS����G1 Continuity����When the two patches meet with tangent (G1) continuity, the reflection lines meet but kink at the patch boundary.�Courtesy of Parametric Technology Corp. © 1995�Data generated by Pro/CDRS����G2 Continuity����With higher orders of continuity (here, G2), the reflection lines are smooth as they cross the patch boundary. This can be quite important in such fields as car design.��v���M���X � À �)��� "€š�€��0¤ª‚l€�‚‚‚ÿ�Courtesy of Parametric Technology Corp. © 1995�Data generated by Pro/CDRS���������„ � À �q† �2������„ �>À �.��� ,€ �€��0¤¨€�†"€���‚ÿ������ç�� À �W �2��� 2€Ï�€��0¤ª‚l€�‚‚€ �‚€�‚‚ÿ����Evaluating Surface Characteristics: Hedgehog Display���Another way to view surface continuity and shape is through hedgehog display. Each "spine" of the hedgehog is a surface normal to the displayed object. The length of the normal varies with the curvature of the surface. Regions of higher curvature are shown by longer surface normals. For this tool to be useful, the parameterization must be carefully chosen to give consistent surface normals on the different surface patches.��6��æ��>À �Ä �P��� n€Ñ�€��0¤ª‚l€�‚‡"€��$€�€�‚‚‚‚‚‡"€��%€�€�‚ÿ���G0 Continuity����When the patches are connected with positional (G0) continuity, the surface normals exhibit significant change in both size and direction at the patch boundary.�Courtesy of Parametric Technology Corp. © 1995�Data generated by Pro/CDRS����G1 Continuity����Where surface patches meet with tangent (G1) continuity, the surface normals are parallel at the boundary but are not of equal length. This indicates that there is a discontinuity in curvature at the boundary.��H����W �ÕÅ �@��� N€�€��0¤ª‚l€�‚‚‚‚‡"€��&€�€�‚‚‚‚ÿ�Courtesy of Parametric Technology Corp. © 1995�Data generated by Pro/CDRS����G2 Continuity����With G2 continuity, the surface normals have equal length and direction across the boundary.�Courtesy of Parametric Technology Corp. © 1995�Data generated by Pro/CDRS���2������Ä �Æ �.��� ,€ �€��0¤¨€�†"€���‚ÿ������Ê��ÕÅ �È �>��� J€—�€��0¤ª‚l€�‚‚€ �‚€�‚‚‚‡"€��'‚‚ÿ����Faceting: The Display of Complex Surfaces���The display of complex surface patches often requires approximating the patches with polygonal facets. Graphics workstations usually accept triangular or quadrilateral facets, which are shaded and displayed by hardware. The resolution of the facets can be controlled to optimize the speed or quality of display.���This image shows the facets of a joystick that will be used to display the part.�Alyn Rockwood��¬���x���Æ �»È �4��� 8€ò�€��0¤ª‚l€�‚‚‡"€��(‚‚‚ÿ����Here, the joystick is fully rendered using Gouraud shading, a standard graphics illumination model.�Alyn Rockwood���2������È �íÈ �.��� ,€ �€��0¤¨€�†"€���‚ÿ����_����»È �LË �K��� d€+�€��0¤ª‚l€�‚‚€ �‚€�‚‡"€��)€�€�€�€�‚ÿ����The Utah Teapot: Texturing and Bump Mapping����Several methods exist to apply texture to a surface by mapping color patterns onto the surface. The preceding figure shows the Utah teapot, a popular benchmark for surfaces.����The teapot is textured by a random polka dot pattern. Bump mapping [Blinn78] perturbs the normal of the surface so that the object appears dimpled when illuminated by standard graphics routines.����Texturing methods greatly extend the application domain for surfaces in computer graphics and animation.��9������íÈ �…Ë �(��� €"�€��2˜¦¨‚l€�‚ÿ�Peter Chambers��*������LË �¯Ë �'��� €�€��0¤ª‚l€�‚ÿ���2������…Ë �áË �.��� ,€ �€��0¤¨€�†"€���‚ÿ����í��ª��¯Ë �ÎÎ �C��� T€W�€��0¤ª‚l€�‚‚€ �‚€�‚‡"€��*€�€�‚ÿ����The Utah Teapot Rendered by Contouring���� This time the Utah teapot is rendered by contouring. This method is also known as "water levels" or "breadslicing." It indicates a set of parallel plane cuts through the surface. Contouring can reveal surface characteristics that are difficult to discern with natural illumination. For example, it indicates which parts of the surface are at the same level, given an orientation of the cutting plane with respect to the object.����Flat areas and singular regions (saddle points and extrema) also become readily evident. Such areas in the teapot are seen on the handle, the spout, the body near the handle, and on the knob of the lid.��8������áË �Ï �(��� € �€��2˜¦¨‚l€�‚ÿ�Alyn Rockwood��*������ÎÎ �0Ï �'��� €�€��0¤ª‚l€�‚ÿ���2������Ï �bÏ �.��� ,€ �€��0¤¨€�†"€���‚ÿ����Î��‹��0Ï �< �C��� T€�€��0¤ª‚l€�‚‚€ �‚€�‚‡"€��+€�€�‚ÿ����Simple Display and Illumination of a Turbine Blade����A simple illumination model, simubÏ �< �q† �lating just diffuse lighting together with highlights, is less than realistic in effect. However, it reveals essential aspects of the object as clearly as a highly detailed rendering.����Transparent surfaces permit viewing of underlying surfaces while maintaining the relationship of the outer surface.��8������bÏ �t �(��� € �€��2˜¦¨‚l€�‚ÿ�Alyn Rockwood��*������< �ž �'��� €�€��0¤ª‚l€�‚ÿ���2������t �Ð �.��� ,€ �€��0¦¨€�†"€���‚ÿ����,������ž �ü �(��� €�€��0¤ª‚l€�‚‚ÿ����8������Ð �4 �'��� €"�€���¤€ �€�‚ÿ�Image Gallery���o����ü �£ �R��� r€?�€��0¤ª‚l€�‚‚‚‡"€��,€�€�‚‚‚‚‚‡"€��-€�€�‚ÿ��The following images show the diversity and effectiveness of modeling with surfaces, especially with good display routines.���Automobile Front End����The combination of CAGD and high-quality rendering algorithms produces considerable realism, allowing for aesthetic consideration of an object.�Courtesy of Parametric Technology Corp. © 1995�Data generated by Lan Zaback, Pro/CDRS, rendered in Pro/PHOTORENDER����Ski Boot����Realistic display techniques allow for product evaluation, marketing, and other commercially useful applications.��D��ñ��4 �ç �S��� t€ç�€��0¤ª‚l€�‚‚‚‚‚‡"€��.€�€�‚‚‚‚‡"€��/€�€�‚ÿ�Courtesy of Intermountain Design, Inc.�Data generated by Pro/CDRS, rendered in Pro/PHOTORENDER�© 1995, Intermountain Desig