CS 284: CAGD 
Lecture #10 -- Mo 9/29, 2003.

Preparation:

Topics: Surface Patches; Subdivision Techniques

Introduction to "General Subdivision (with modification)" Techniques

• Conceptual introduction via iterated refinement

- - carving a rounded object from wood.
- - cutting a rounded shape from paper by "repeated corner cutting."
- - adding extra "bulges" on the segments of a linear spline.
- - re-adjusting the mid-points of all control segments.
- - averaging vertices among their neighbors.
 
• Key points about useful subdivision schemes:

- - 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.
- - the subdivision approach failed with the circle splines, because there is no affine invariance.
 
• Special case study: 
  - - interpolating four points with a cubic polynomial, to find new mid point for subdivision.
  - - this is the same as averaging two quadratic interpolants through three points each.

Formal Subdivision

• Iterated affine transformations (Warren+Weimer, p 9)

- - Defined by the transformation of three non-collinear points ==> Barycentric coordinates.
• Fractals: a set of contractive transformations. Examples:

- - Sierpinski triangle
- - Koch snowflake
- - Bezier curve (iterated mid-point evaluation) based on de Casteljau
• Subdivision as the limit of of an increasingly faceted sequence of polygons (WW p 19)

- - each is related to successor by a simple linear transformation.
• Subdivision as a multiresolution rendering algorithm that generates increasingly dense plots

- - by taking convex combinations of points used in previous plots (WW p 24).
• Importance of the subdivision matrix (WW p 25).

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 (p140)
• Patch subdivision, degree elevation ... All still works as in the 1D case!
• Putting Bezier patches together with G1 or better continuity becomes difficult and painful.
• If you want a high degree of continuity, consider the approximating B-spline surfaces:
• Bicubic and biquintic B-spline patches
• Rectangular B-spline surfaces
• Comparison, trade-offs between Bezier and B-Spline surfaces

Next time: 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!
• The classical subdivision surface by Catmull & Clark = foundation for all modern subdivision algorithms.

Take-home Quiz on Curves.

DUE: WEDNESDAY, October 1, 9:10am

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Next Homework Assignment: (to be done individually)

Experimenting with Curve Subdivision Schemes

• For instance, form an approximating curve by repeatedly chopping off the corners of the control polygon (as demonstrated in class).
• To make an interpolating curve, repeatedly subdivide the polygon edges by introducing a new vertex somewhere near its middle (or perhaps ratioed by the lengths of the adjacent segments of the control polygon -- or by the cube root of that ratio), then move that point to a place that will help to make an overall smooth curve (perhaps place on best-fitting circle).

Also submit a window capture for each applet, showing an interesting case of a control polygon with rather irregularly spaced points and sharp angles in the control polygon. Add to each one a description of how you chose to place the new subdivision vertices.

For more information see the instructional pages.

DUE: WED 10/8/03, 9:10am.

On line: Follow submission instructions on the instructional pages.

Hand in: Pictures of two interesting curves and descriptions of your subdivision schemes.

Next Reading Assignment: