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

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

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

Surfaces

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

• Do in "u" and in "v" directions what we have learned in "t" direction ...
• Simplest case:  Bilinear Bézier patch = Coons Patch
• Cubic tensor-product 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 B-spline surfaces:
• Bicubic and biquintic B-spline patches
• Rectangular uniform B-spline surfaces
• Typically same degree in both parameter directions -- but this is not required.
  
• Comparison, trade-offs between Bézier and B-Spline surfaces
• B-splines guarantee continuity, but have less direct handles.
• Hardware support (e.g. for rendering) exists primarily for Bézier .
• May do your design as B-splines, then represent data as Bézier patches (use blossoming to get control points).
  (This is what Raph Levien does with his Euler-spiral splines).
  

Triangular Surfaces Patches

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

Concept of Recursive Subdivision 

• 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 half-way 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