CS 284: CAGD 
Lecture #6 -- Mo 9/15, 2003.

Preparation:

Main Lecture Topic

An interpolation task (see B+B+B, Chapter 3): Comparison of various approaches:

• Cubic Hermite interpolation [see B+B+B, Chapter 3]:
• Continuity =?; DoF = ?; Extent of support = ?.
• Non-uniform Lagrange interpolation:

Continuity = ?; DoF = ?; Extent of support = ?.
• Linear positional circle interpolation [Wenz]:

Continuity = ?; DoF = ?; Extent of support = ?.
• Trigonometric positional circle interpolation [Szilvasi-Nagi]:

Continuity = ?; DoF = ?; Extent of support = ?.
• Linear angle-interpolated circle splines [Sequin et al]:

Continuity = ?; DoF = ?; Extent of support = ?.
• Trigonometric angle-interpolated circle splines, reparameterized for C2-continuity [Sequin et al]:

Continuity = ?; DoF = ?; Extent of support = ?.
• Your stitched-together composite Bezier curve:

when G¹-continuity is OK: --> DoF = ?; Extent of support = ?.
when C¹-continuity is required: --> DoF = ?; Extent of support = ?.

Some Comments on Bezier-stitching Homework

• Not as easy as it first seems -- everybod showed good effort, -- not everybody succeeded completely.
• First crucial step is to get the tangent direction right.
• Line parallel to P₀ - P₂ does not lead to great solutions, but it can maintain affine invariance.
• Using a tangent perpendicular to the angle-bisector in the control polygon, gives better results.
• For even better results, some more weight should be given to the shorter segment to define tangent direction;

it is an open-ended question how this should be done for best results ...
(perhaps divide the turning angle between the control polygon segments proportional to the ratio of the segment lengths).
• For G¹-continuity the Bezier control points adjacent to the joint need to lie both on this tangent line.
• They can be at un-equal distances; C¹-continuity does NOT lead to better looking shapes, it just adds one more constraint!
• The distances should be scaled (somewhat) with the length of the underlying control segment.
• The construction should be structurally symmetrical, i.e., use essentially the same formula for both inner control points.

Compare your solution with various other approaches:

• Trigonometric angle-interpolated circle splines, reparameterized for C²-continuity [Sequin et al]
• Linear angle-interpolated circle splines [Sequin et al]
• Trigonometric positional circle interpolation [Szilvasi-Nagi]
• Linear positional circle interpolation [Wenz]
• Non-uniform Lagrange interpolation.

Bottom Line: Polynomials are not the only possible primitives!

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Minimum-Variation Curves

• Choose any suitable representation
• Minimize the arc-length integral over square of derivative of curvature, by using the un-constraint DoF.
• One possible approach uses quintic Hermite splines [Moreton and Sequin],

and a good heuristic to set the initial tangent directions (1st derivatives) and curvatures (2nd derivatives) at the interpolation points.

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Circle Splines

• SIGGRAPH'03 Presentation
• Kiha's demos: use Explorer!

Current Homework Assignment: (to be done individually)

New DUE DATE: Friday 9/19/03, noon.

On line:
  Set the initial values for the sliders to the proper values, so that when we execute your program,
 we will see a nice "prismatic" sweep of a 3-pointed star along your best composite Bezier curve.
 (You may want to change some of the parameters from the values that you gave them in pa1.)
 Capture a hardcopy print-out of this sweep.
  Put your SLIDE file in the proper place {see instructional page} (same as with your last assignment);
  and also mail a copy of your final modified pa2.slf file to jordans@cs.berkeley.edu

Hand in at the beginning of class:
  A hardcopy print-out of your sweep;
  and a page or two of text, answering the questions raised above.

Reading Assignment: