CS 284: CAGD 
Lecture #9 -- Tu 9/26, 2006.

Preparation:

Creative Thinking

Topic:  B-Splines (cont.)

Reviewing some key aspects of B-Splines

• B-Spline Basis Functions
  
  • Strongly overlapping control domains leads to built-in smoothness.
• Periodic (closed) B-Spline Curves (p 105)
• End-around re-use of control points.
• B-splines of different degrees (Applet on p 102)
• How many control segments does it take to make the first curve segment appear ?
• Note that quadratic B-spline touches control polygon -- Why ?
• Use of B-splines:
• What can you do with a given number of segments ? 
• How many segments does it take to make a knotted 3D space curve ?

Vertex Multiplicities

• Piling deBoor points on top of one another
• Effects on parametrization ?
• Effects on basis functions ?
• Effects on the B-spline curve ?
• Experiment with the interactive display panels shown in the book on page 102.

Extend a cubic curve by 6 more points and then move the de Boor control points
to study what happens to the B-spline curve when you:
1. double up two de Boor points at the end ?
2. tripple a de Boor end-point ?
3. give a de Boor end-point a multiplicity of four ?
4. double up two internal de Boor points ?
5. make a tripple internal de Boor point ?
6. give an internal de Boor point a multiplicity of four ?

Non-uniform B-Splines

• Changing the Knot Values
  
• can assume arbitrary, monotonically ordered t-values
• does this affect only the parametrization, or also the shape of the curve ?
• study their influence with applet on p105.
• Effect on B-Spline Curve (Applet on p106)
• What is the effect on reducing the knot interval ? 
• What happens when we double-up knots 1 and 2, or knots 3 and 4 (Applet on p 107)
• Effect on Basis Functions (Applet on p110)
• Study effect of shifting knots for degree 1 ... 4 basis functions.
• How far does the effect of a changed knot value spread ?

Knot Multiplicities

• Gives additional design freedom
• Effects on the basis functions (BBB p162-166)
• Effects on the B-spline (BBB p167-172)
  ==> see handout.

Knot Insertion (Curve Refinement)

• New knots can be inserted anywhere at will;
• they can be used for curve refinement or for splitting curves at desired locations.
  
• Review of knot insertion and B-spline subdivision
• The general approach using Blossoming for an arbitrary value for t.
• Degree elevation of a B-spline segment
• Using Blossoming and detour via Bezier control points.
• Review of some facts about behavior of knots

Topics: Surface Patches

From Curves to Surfaces Patches

• Do in "u" and in "v" directions what we have learned in "t" direction ...
• Bilinear Bezier patch = Coons Patch
• Cubic tensor-product Bezier patch
• Symmetry in u,v: interchange roles of "rails" and "curves"
• Biquintic Bezier patch
• DeCasteljau evaluation of tensor product patches (p140)
• Patch subdivision, degree elevation ... All still works as in the 1D case!
• Putting Bezier patches together with G1 or better continuity becomes difficult and painful.
• If you want a high degree of continuity, consider the approximating B-spline surfaces:
• Bicubic and biquintic B-spline patches
• Rectangular B-spline surfaces
• Comparison, trade-offs between Bezier and B-Spline surfaces
  

Triangular Surfaces Patches

Reading Assignment:

Current Homework Assignment: Construct a Parameterized Bell

Create a SLIDE file with all parameters set to their preferred values.
Capture your design pictorially using the screen saver.

• E-mail to me your SLF-file and a captured picture (JPG, GIF, PNG)
  

DUE: Thu. 9/28/2006, 2:10pm.