CS 284: CAGD
Lecture #9 -- We 9/27, 2000.

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Preparation:

B+B+B: Effect of Knot Multiplicities
Rockwood: pp 133-142 (Spline Surface Patches)

Review

Some facts about behavior of knots

Effect of Knot Multiplicities

Gives additional design freedom
==> see handout.

Effects on the basis functions (BBB p162-166)
Effects on the B-spline (BBB p167-172)

Effect of Vertex Multiplicities

(Piling deBoor points on top of one another)

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:

double up two de Boor points at the end ?
tripple a de Boor end-point ?
give a de Boor end-point a multiplicity of four ?
double up two internal de Boor points ?
make a tripple internal de Boor point ?
give an internal de Boor a multiplicity of four ?

Study:

Effects on the B-spline geometry
Effects on parametrization

Topics: Surface Patches; Subdivision Techniques

Introduction to "General Subdivision (with modification)" Techniques

Conceptual introduction
- - carving a rounded object from wood.
- - cutting a rounded shape from paper by "repeated corner cutting."
The classical subdivision surfaces by "Catmull & Clark" and by "Doo & Sabin"
== foundation for all modern subdivision algorithms.

From Curves to Surfaces Patches

Do in "u" and in "v" directions what we have learned in "t" direction ...
Cubic tensor-product Bezier patch
Biquintic Bezier patch
Biquintic B-spline patch
Symmetry in u,v
Rectangular B-spline surfaces
Comparison, trade-offs between Bezier and B-Spline surfaces
--- not yet done ---
DeCasteljau evaluation of tensor product patches
Patch subdivision
Degree elevation

New Homework Assignment: (to be done individually)

Experimenting with Curve Subdivision Schemes

Given a sequence of points that define a control polygon, form smooth interpolating and approximating curves by some subdivision schemes of your own design.

For instance, form an approximating curve by repeatedly chopping off the corners of the control polygon (as demonstrated in class).
To make an interpolating curve, repeatedly subdivide the polygon edges by introducing a new vertex somewhere near its middle (or perhaps ratioed by the lengths of the adjacent segments of the control polygon -- or by the cube root of that ratio), then move that point to a place that will help to make an overall smooth curve (perhaps place on best-fitting circle).

Build those exploratory subdivision routines on top of the Java applet provided in the last assignment. We have provided the framework of last weeks Java code, called pa4, in which we have stripped out all Bezier machinery, but left you with the drawing/editing, and display functionalities. Add to that your new curve drawing routines based on subdivision. Place your new demonstration applet in your instructional NT accounts in a directory hw/pa4/.

Also submit a window capture for each applet, showing an interesting case of a control polygon with rather irregularly spaced points and sharp angles. Add to each one a description of how you chose to place the new subdivision vertices.

For more information see the instructional pages.

DUE: WED 10/4/00, 9:10am.

On line: Follow submission instructions on the instructional pages.

Hand in: Pictures of two interesting curves and descriptions of your subdivision schemes.

Next Reading Assignment:

Handout: Two papers by "Catmull and Clark" and by "Doo & Sabin"