CS 284: CAGD 
Lecture #9 -- Thu 9/20, 2012.

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Preparation: READ: Rockwood Chapter 8: pp 133-151: Surfaces

Warm-up Exercise: How to design a nice geometric mug with a well-integrated handle?


We spent 1/3 of the course on just curves (= 1-manifolds) {except for Sweeps}.
The good thing is that the math and all the insights gained apply in an almost identical manner to to 2-manifolds (patches, surfaces)!

A quick preview:

Bézier Patches: Remember how hard it was to stitch together Bézier curve segments with higher order continuity
-- for Bézier patches this is even harder!

B-spline Patches: They come to the rescue. They are only approximating, but provide automatically good continuity.
If we want good continuity and good geometric control, then we also need to allow manipulation the knot values.
If we want to make exact circular constructs, we also need to make the basis functions rational. ==> NURBS.
If we want to make conveniently shapes of higher genus, then fitting quadrilateral B-spline patches together becomes problematic.
Subdivision schemes have been devised to make this process easier.

From Curves to Surfaces Patches

Triangular Surfaces Patches

We can also deal with triangular patches, but need a different interpolation scheme:
Barycentric coordinates: three numbers, but with the constraint that they must sum to a constant value.
DeCasteljau evaluation technique can also be applied to triangular patches.

Topological Limitations of the B-spline Control Mesh

Concept of Recursive Subdivision 

(This will be a focus for the next few weeks.)

Homework Assignments for Tue Sept. 25, 2012

Everybody needs to read:
The seminal paper by E. Catmull and J. Clark:"Recursively generated B-spline surfaces on arbitrary topological meshes"
Paper by D. Doo and M. Sabin: "Behavior of recursive division surfaces near extraordinary points"
3.)  Chapters 2 and 3 from: C. Loop, "Smooth Subdivision Surfaces Based on Triangles" --  Some errata found in this thesis.

In addition: prepare a 10- to 15-minute, in-class, formal presentation (with slides, PPT ...) as follows:
Paper # 1.) -- Laura and Eric
Paper # 2.) -- Michael
Paper # 3.) -- Soham and Brandon

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