CS 284: CAGD
Lecture #9  Thu 9/20, 2012.
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Preparation: READ: Rockwood Chapter 8: pp 133151: Surfaces
Surfaces
We spent 1/3 of the course on just curves (= 1manifolds) {except for Sweeps}.
The good thing is that the math and all the insights gained apply in an
almost identical manner to to 2manifolds (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!
Bspline 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 Bspline patches together becomes problematic.
Subdivision schemes have been devised to make this process easier.
From Curves to Surfaces Patches

Do in "u" and in "v" directions what we have learned in "t" direction ...
 Simplest case: Bilinear Bézier patch = Coons Patch

Cubic
tensorproduct Bezier patch

Symmetry in u,v: interchange roles of "rails" and "curves"

Biquintic
Bezier patch

DeCasteljau evaluation of tensor product patches (p138)
 use DeCasteljau on "control rails" then on "ribs".

Patch subdivision (p140), degree elevation ... All still work as in the 1D case!
 subdivide one direction of control polygons to get new mid patch boundary, ...
 use convex hull of subdivided control polygons for intersection tests, clipping, ...

Putting Bézier patches together with G1 or better continuity is difficult
and tedious.

If you want a high degree of continuity, consider the approximating Bspline
surfaces:

Bicubic
and biquintic Bspline patches

Rectangular uniform Bspline surfaces
 Typically same degree in both parameter directions  but this is not required.

Comparison, tradeoffs between Bézier and BSpline surfaces
 Bsplines guarantee continuity, but have less direct handles.
 Hardware support (e.g. for rendering) exists primarily for Bézier .
 May do your design as Bsplines, then represent data as Bézier patches (use blossoming to get control points).
(This is what Raph Levien does with his Eulerspiral splines).
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 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
Concept of Recursive Subdivision
(This will be a focus for the next few weeks.)

Conceptual introduction via iterated refinement
 Cutting a rounded shape from paper by "repeated corner cutting." (DEMO)
 Carving a rounded object from wood or styrofoam by cutting away edges. (DEMO)
 Adding extra "bulges" on the segments of a linear spline (as in a fractal construction, e.g. Koch snowflake curve).
 Calculating intermediate data points halfway between the given interpolation points (not the whole curve segment).
 Smoothing by (weighted) averaging vertices among their neighbors.
 A mixture of several of the above techniques.
Homework Assignments for Tue Sept. 25, 2012
Everybody needs to read:
1.) The seminal paper by E. Catmull and J. Clark:"Recursively generated Bspline surfaces on arbitrary topological meshes"
2.) 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 15minute, inclass, formal presentation (with slides, PPT ...) as follows:
Paper # 1.)  Laura and Eric
Paper # 2.)  Michael
Paper # 3.)  Soham and Brandon
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