CS 284: CAGD
Lecture #11 -- We 10/4, 2000.

Preparation:

Topics: Non-rectangular Patches, Subdivision Surfaces

A Quadratic Subdivision Scheme

• See the end of the Doo-Sabin paper.
• Compare the subdivision patterns for the two schemes.

Using Rectangular Patches to Make Shapes of Desired Geometry
(Your new homework assignment)

• Let's build something from the "tensor-product" patches we have been discussing.
• The key is to place the control points properly.
• Remember what you did/learned for curves in the first three assignments.
• The points will roughly outline the overall shape of the object you want to make.
• Example: to make something roughly cylindrical
• Example: to make something roughly toroidal
• Your new assignment: make a Moebius band and a Klein bottle
  -- or, if you want to work in pairs: make a Cross-cap.
• What is a Moebius band ?
  -- a closed ribbon with an odd number of half-twists.
• What is a Klein bottle ? It comes in two varieties:
  -- the classical model of the Klein bottle, with many physical examples that you can buy,
  -- the figure-8 shape Klein bottle.
• What is a cross-cap ?
  -- it is a finite immersed model of the projective plane.

Discussion of Results of Current Homework

• Which scheme is easier to implement for robust behavior ?
• What difficulties have you encountered ?
• What remedies have you tried ?
• Any inspiration from the Zorin paper ?

Interpolating Subdivision Surface for Triangulated Control Polyhedra

• Discussion of the Zorin paper

Homework Assignment #5: (to be done individually -- or in pairs )

Make Smooth Closed Surfaces in SLIDE

For the Moebius band, place the control points for a m-by-n bezier patch so that you get a ribbon that closes on itself with C1 continuity. Don't just make a boring "ruled surface" (i.e., a bent "flat" ribbon with n=2), or something that could just as easily be done with a sweep. Choose n>=4, and m>=7. Feel free to vary the width of the ribbon. Add some texture to the patch, so that the parametrization becomes more visible.
If you want a challenge -- make a Moebius strip with no self-intersections that has its edge in the shape of a single planar circle ! (This is known as "the Goblet" -- it looks like a crosscap in which one of the the seams along the double intersection lines has popped open - into a circle). Two of these "Goblet" Moebius bands can then be joined mouth-to-mouth to form a Klein bottle.

For the Klein bottle, place the control points -- manually or procedurally -- to define a m-by-n array of quadrilateral B-spline patches that together form a C2-continuous Klein bottle. You will need to properly identify which points will get reused along the seams to obtain the desired smooth closure. You can construct either type of Klein bottle (standard of figure-8).

For this assignment, you can elect to work as a pair. In this case, rather than constructing a Klein bottle, you will need to construct a cross-cap with good parametrization. A cross cap is NOT a pinched-off torus ! There should be no parameter lines that get crunched into oblivion. Use the parametrization indicated on the page above.

If you plan to place some or all of your control points procedurally within a tcl_init block in your SLIDE program, then here is an example file that shows how the array of control points for a Bezier patch can be generated with tcl.

DUE: Thursday 10/12/00, midnight.

On line: put your SLIDE/tcl code into U:\cs284\hw\pa5\

Hand in: Pictures of your two surfaces and a description of how you constructed/placed the control points for them.

Next Reading Assignment:

Monday October 9:
We will have an in-class quiz about the material covered so far

The test will be closed book --
however, you will be allowed to bring one 8.5" by 11", double-sided sheet of notes to the test.