CS 284: CAGD 
Lecture #9 -- Mon 9/28, 2009.

Preparation:

Warm-up Problem: How to model a bell

Possible approaches to composing a Bell shape

Topic: Surface Patches

From Curves to Surfaces Patches

• Do in "u" and in "v" directions what we have learned in "t" direction ...
• Bilinear Bezier 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 Bezier 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 Bezier and B-Spline surfaces
• B-splines guarantee continuity, but have less direct handles.
• Hardware support (e.g. for rendering) exists primarily for Bezier.
• May do your design as B-splines, then represent data as Bezier patches (use blossoming to get control points).
  (This is what Raph Levien does with his Euler-spiral splines).
  

Triangular Surfaces Patches

Topic: Subdivision

Introduction to "General Subdivision (with modification)" Techniques

• 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)
  
• Smoothing by (weighted) averaging vertices among their neighbors.
  
• Calculating intermediate data points half-way between the given interpolation points (not the whole curve segment).
• Adding extra "bulges" on the segments of a linear spline (as in a fractal construction, e.g. Koch snowflake curve).
  
  
• Special case study: Cubic interpolation: 
  Interpolating four points with a cubic polynomial to find a new mid point for subdivision
   is the same as averaging two quadratic interpolants through three points each.
 
• Key points about useful subdivision schemes:
• There are two components to any subdivision scheme, a topological 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.
  
  
• Subdivision of cubic B-spline
• Calculate a control polygon with twice as many control points.
• Doing this in 2-dimensions is the basic idea behind the Catmull-Clark subdivision scheme for surfaces.
  

Reading Assignment:

Current Homework Assignments:

Construct a Parameterized Goblet

Create a SLIDE file with all parameters set to their preferred values.
Capture your design pictorially using the screen saver.

• E-mail to me your SLF-file and a captured picture (in JPG, GIF, or PNG)
  

DUE:  Wed. 9/30/2006, 10:40am.