CS 284: CAGD 
Lecture #13 -- We 10/8, 2003.

Preparation:

Discussion of Homework: Experimenting with Curve Subdivision Schemes

The Design Process

Topic: The Subdivision Process (cont.)

Review: Doo, Sabin Paper

• Extension of Chaikin's Corner Cutting algorithm (1974) to surfaces.
• For rectangular quad meshes results in quadratic B-spline surface.
• Careful analysis and treatment of irregular points.

Subdivision Masks for Surfaces

• Tensor Product Splines
• E. Catmull and J. Clark (1978)
• Cubic quad stencil
• Weight masks for new vertices
• Doo and Sabin (1978)
• Quadratic quad stencil
• Weight masks for new vertices
• Triangle Splines
• C. Loop (1987)
• Quartic triangle stencil
• Weight masks for new vertices, regular case
• Weight masks for new vertices, generalization
• Different weights introduced by Warren
• Triangle spline as the shadow of a cube
• Subdivision of shadow cube

Loop Subdivision

• Construction (Loop thesis, Chapter 3)
• Analysis (Loop thesis, Chapter 4)

Warren & Weimer, Chapter 2

• 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.

Next Reading Assignment: