CS 284: CAGD 
Lecture #12 -- Mo 10/6, 2003.

Preparation:

Discussion of Take-Home Quiz #1

Topic: The Subdivision Process

Review of Catmull, Clark Paper

• A generalization of the cubic B-spline scheme to arbitrary topologies ( based on "vertex averaging").
• Some adhoc rules about what to do at non-valence-4 vertices.
• The evolution of the irregular regions in subsequent subdivision generations.

Doo, Sabin Paper

• A subdivison scheme based on quadratic B-splines ( similar to "corner cutting")
• A more detailed analysis what happens at irregular points
• Use of matrices in the subdivision process
• Analysis of the eigenstructure to understand the behavior of the limit surface.
• Frequency analysis of the surface piece around a point

Warren & Weimer, Chapter 2

• Pg. 27: A high-level overview. Study this chapter until this makes sense.
• Pg. 28: A nice review/summary of many of the things we have discussed in class.
• Ch. 2.1.1: B-Spline Basis Functions defined by repeated integration / convolution.
• Ch. 2.1.2: Refinement Relations: Each basis function is also a linear sum of compressed copies of itself.
• e.g. order m=1, piecewise constant: Eqn. (2.5) and Fig. 2.6
• Generalization: Theorem 2.1
• Ch. 2.1.3: More on subdivision expressed in matrix form
• The B-Spline weighting coefficients appear as column vector segments, staggered with a vertical shift of 2, in the matrix.
• Ch. 2.2.1-2: B-Spline Basis Functions defined as cross sectional volumes of hypercubes.
• Ch. 2.2.3: Subdivision (refinement) scheme resulting from above definition.
• Ch. 2.2.4: Applying this scheme to two-manifolds (to make surfaces rather than curves).
• Ch. 2.3.1: B-Splines and Box Splines defined as piecewise polynomials.

Next Homework Assignment: (to be done individually)

Design the Control Mesh for a Genus-4 (minimum energy) Surface.

• WED 10/8: Hand in a sketch of the rough geometry of the object that you plan to construct,

and a paragraph that outlines your plan for constructing the actual control mesh.
• MON 10/13: Hand in a printout of a simple symmetrical mesh of quadrilaterals.

List the control parameters that you have for your mesh.
• WED 10/15: Complete assignment due. Hand in a printout of a smooth Catmull-Clark surface;

and submit your SLIDE file electronically.

Current Homework Assignment: (to be done individually)

Experimenting with Curve Subdivision Schemes

• 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).

Also submit a window capture for each applet, showing an interesting case of a control polygon with rather irregularly spaced points and sharp angles in the control polygon. 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 10/8/03, 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: