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
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 noncollinear 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 midpoint evaluation) based on de Casteljau (handout WW pp 1518)

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 Bspline 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 uvcoordinates 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 Bspline 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
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 online !)
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 reoriented 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 "onionlike"
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 email 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 finetune your mesh.
PHASE III DUE: Tu. 10/10/2006, Noon!:
 The complete, final, subdivided, and thickened surface (I have added an offsetsurface 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.
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