CS 284: CAGD 
Lecture #12 -- Th 10/5, 2006.

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Chapters 2 and 3 from: C. Loop, "Smooth Subdivision Surfaces Based on Triangles"
This is also a nice review of B-spline subdivision!

Warm-Up:  Devise your own subdivision scheme !

Topic: Subdivision (cont.)

Subdivision Schemes for Surfaces

Tensor-product patches, e.g., cubic tensor-product Bezier patches;
readily do in the "u" and in "v" directions what we have learned about curves.

We can also use with triangular patches, but need a different interpolation scheme:
Barycentric coordinates: three numbers, but with constraint that they must sum to 1.0.
DeCasteljau evaluation technique can also be applied to triangular patches!

Evaluation / Analysis of Subdivision Schemes

How do we know whether a particular interpolation or subdivision scheme is any good ?

Testing / Evaluation by Visual Inspection

Formal Analysis of Blending / Subdivision Methods

Reading Assignments:

Zorin et al: "Interpolating Subdivision Meshes with Arbitrary Topology"

Chapters 4 and 5 from: C. Loop, "Smooth Subdivision Surfaces Based on Triangles"

New Homework Assignment: Devise an Interpolating Subdivision Scheme for Quad-Meshes.

Continue with the Warm-up Exercise in the same five groups.
Exchange phone numbers, e-mails, available time slots ...  NOW.
Meet as a group at least twice between now and  Tuesday to discuss this task.
As a team, e-mail me an interim report (less than 1 page) describing your scheme and its preliminary evaluation.

DUE:  TUESDAY 10/10/2006, Noon!:

Last Phase of current Homework Assignment: Create a Subdivision Surface

Create a doubly or triply spiralling surface.
Similar to the Creative Thinking Exercise on Koch's Snowflake Curve, you should try to find a surface in 3D space that is inspired by a logarithmic spiral in the plane.
However, you must not just extrude a logarithmic spiral in a direction perpendicular to the spiral plane. You should create a surface that shows some spiralling cut lines when cut in as many different orientaions as possible. The surface will probably have to have some (spiralling?) edges -- which is good, because this will define some windows through which one can look inwards to the inner parts of the surface.

Keep your surface modular. Model as little as absolutely needed; then put multiple copies suitably re-oriented together to make the complete surface.
In addition to exploiting symmetry at one (spherical) level of the surface, you should then extend the surface inward or outward by simply making suitably scaled copies of one layer of that "onion-like" assembly. The use of one or two parameters to optimize the look of the surface is encouraged.

I have put some SLIDE starting file "spiral.slf" into the CODE directory. It has most of the basic elements that you will need to build such a surface and shows how to do mirroring, scaled instanciation, and subdivision. I also have included some token parameters so you can get started with something that already works and then do incremental modifications.

Here are also some (hopefully inspirational) images!

PHASE I --DUE: Tu. 10/3/2006, 2:10pm:

Plan the topology (connectivity and rims)
of your surface to get the desired spiral patterns locked in. Give me something by Tuesday at the latest that allows me to give you feedback whether you are on a good track. You can either give me a paper and pencil sketch, or a rudimentary slide file that shows the basic geometry, even though not all pieces fit together seamlessly yet. Feel free to e-mail me images or SLF files before Tuesday.

PHASE II --DUE: Th. 10/5/2006, Noon!:

PHASE III --DUE: Tu. 10/10/2006, Noon!:

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