CS 284: CAGD
Lecture #10  Wed 9/30, 2009.
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Preparation:
Read the seminal paper by Catmull and Clark on subdivision surfaces
Topic: Subdivision (cont.)
Topological Limitations of the Bspline 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 uvcoordinates 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 Bspline scheme

Can handle vertices with valences different from 4

Can handle meshes with facets other than quadrilaterals

One subdivision iteration calculates:
 new vertices at the centers of all faces,
 new vertices "below" the centers of all edges,
 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
valence4 vertices,

In these ever more dominating "regular" regions, the surface will approach
a Bspline.
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:
Paper by Doo and Sabin on subdivision surfaces
Chapters 2 and 3 from:
C.
Loop, "Smooth Subdivision Surfaces Based on Triangles"
This is also a nice review of Bspline subdivision!
Some
errata found in this thesis.
New Homework Assignment:
Design the Control Mesh for a Smooth Genus4 Surface.
The goal is to design a highlysymmetrical control mesh for a closed genus4
subdivision surface
based on triangles
(which could be later used for experiments in surfaceenergy minimization
studies).
Following an iterative
design process, we will do this in stages:
 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)
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