CS 284: CAGD
Lecture #11 -- Tu 10/3, 2006.

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Preparation:

Paper by Doo and Sabin on Behavior of subdivision surfaces near extraordinary points

Warm-Up Discussion: How to design a "Multi-Spiral Surface"?

How to get started: ... with pipe cleaners: Net1 ... Net2 ... with paper models ...
Design phases.

Topics: Subdivision (cont.)

Conceptual Bridges to Splines, Fractals, and Affine Transformations

Iterated affine transformations (Warren+Weimer, p 9)

Defined by the transformation of three non-collinear points to three other ones;
everything is then referenced to these three points with barycentric coordinates.

Fractals: a set of contractive transformations (distance between any two points shrinks). Examples:

Sierpinski triangle
Koch snowflake
Bezier curve (iterated mid-point evaluation) based on de Casteljau (handout WW pp 15-18)

Subdivision as the limit 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) -- becomes clearer later.

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

Doo, Sabin Paper: Focus on a quadratic subdivision surface

Extension of Chaikin's Corner Cutting algorithm (1974) to surfaces.
For rectangular quad meshes this results in quadratic B-spline surface.
Gain an understanding of the role of the subdivision matrix and its eigenvalues.
Careful analysis and treatment of irregular points and convergence behavior around it.
Discrete Fourier transform of the oscillations of the rings of neighboring vertices.

Subdivision Masks for Surfaces

Tensor Product Splines
E. Catmull and J. Clark (1978)

Doo and Sabin (1978)

Triangle Splines (C. Loop, 1987) next ...

Reading Assignments:

Chapter 2 and 3 from:
C. Loop, "Smooth Subdivision Surfaces Based on Triangles", Master's thesis, University of Utah, Department of Mathematics, 1987.
(! Large Document -- perhaps read on-line !)

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.

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!:

Send me an SLF file showing your control mesh for the critical module;
Include sliders for the control parameters that you use to fine-tune your mesh.

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

The complete, final, subdivided, and thickened surface (I have added an offset-surface module to "spiral.slf");
send me an SLF file and captured image showing best view.

NICE JOB ON THE LAST ASSIGNMENT !! See my Bell Selection.