CS 284: CAGD 
Lecture #10 -- Wed 9/30, 2009.

Preparation:

Warm-up Problem: Devise an Interpolating Subdivision Surface

Topic: Subdivision (cont.) 

Topological Limitations of the B-spline Control Mesh

• A rectilinear mesh of quadrilaterals can be nicely mapped onto a torus.
• It cannot be mapped onto a sphere without either
• crunching the u-v-coordinates together at the N- and S- poles, or
• or introducing vertices with valences different from 4.
• It cannot be mapped nicely onto surfaces of genus higher than 1.
• For these kinds of surfaces, we need a different, more general scheme ==> Subdivision surfaces!
• Remarks on the Topology of Surfaces

The Classical Subdivision Surfaces by Catmull & Clark

• This is a foundation for most modern subdivision algorithms
  
• A generalization of the B-spline scheme
• Can handle vertices with valences different from 4
• Can handle meshes with facets other than quadrilaterals
• One subdivision iteration calculates:
1. new vertices at the centers of all faces, 
2. new vertices "below" the centers of all edges, 
3. a new vertex position for each old vertex.
   
(This could be described as a matrix transformation on all the old vertices.)
• After the first iteration step, there will only be quadrilateral mesh facets;
• and from then on, there will be a constant number of extraordinary points:
• vertices of valence <>4 for each such original vertex.
• vertices of valence <>4 for each non quadrilateral mesh in the original control polyhedron.
• After the second iteration, any facet contains at most one extraordinary vertex.
• The extraordinary points will not disappear, but they will become more and more "isolated",
• being surrounded by ever shrinking irregular regions,
• and being separated by more and more quadrilateral meshes joining in valence-4 vertices,
• In these ever more dominating "regular" regions, the surface will approach a B-spline.

Another Subdivision Surface

• A more detailed analysis what happens at irregular points is in the paper by Doo and Sabin.
• this paper also introduces matrices into the subdivision process,
• and the analysis of the eigenstructure to understand the behavior of the limit surface.

Key points about useful subdivision schemes:

• There are two components to any subdivision scheme, a topological one and a geometrical one:
• topology:  In a fixed way split the parametric domain of an edge or a face;
• geometry:  Move some of the old and newly created vertices to new locations (that promise to yield a smoother shape).
• The number of points (line segments) must grow at a geometrical rate with each generation.
• The newly introduced points should have a smoothing effect and converge towards a  limit function.
• This can typically be achieved with affine mapping schemes described with a subdivision matrix.
• The infinite application of this matrix then leads directly to a point on the curve or surface.
  
  

Nice Work on the Goblet Assignment!

New Reading Assignments:

New Homework Assignment:

Design the Control Mesh for a Smooth Genus-4 Surface.

• MON 10/5:  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 10/12:  Complete assignment due. Hand in a printout of a smooth Loop surface;
  
and submit your SLIDE file electronically.

Paper Presentation Assignments (for the future)