# CS 284: CAGD  Lecture #16 -- Wed 10/21, 2009.

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## Warm-up: Discuss these questions with your neighbors:

1. What are the key differences between the way that Wei and Levoy put texture on a surface
and the way that you are supposed to do it in the current programming Assignment ?

2. Could you modify the Wei and Levoy scheme so that
the result maintains the full symmetry of your genus-4 object ?

# More on Subdivision Data Structures and Texturing

## Texture-Mapped Subdivision Surfaces of Arbitrary Genus

The construction of a subdivision surface starts with a simple polyhedron, which is then iteratively refined and smoothed by subdivision.
The texture coordinates applied to the original polyhedron are simply subdivided in the proportion of the topological splits executed.
Problems may arise in assigning texture coordinates, if the surface is not a simple cylindical or toroidal domain.
Whenever the genus of a surface is different from 1, then you cannot regularly tile this surface with quad tile with all valence-4 vertices.
A cube has eight valence-3 vertices, and one geometrical vertex will have to carry different texture coordinates for different faces,
(thus it might be better to carry the texture coordinates with each face, rather than on a shared vertex).

simple cube; but the "cubist" fish shape will extend beyond a single face and overlap into two adjacent faces.

Thus the "fish" texture pattern that get's cut out by any cube face is missing the nose and the tail, but gets those two pieces stuck in from the two sides.
Since these pieces belong to some other fish, they be be of different colors; different cube faces may have different combinations of colors.
Thus, multiple copies of one B&W texture outline, filled with different colors, are needed for the different sides of the cube.
On the cube itself, the texture coordinates will then have to be rotated so that a seemless connection between the different patterns and colors occurs.
In some cases, some of the tiles may also have to be mirrored! (
See the actual texture coordinates used).

Here is another textured example of genus 2.  --  And another one of genus 3.

More complex examples of a genus 5 surface are discussed here.

Here are some actual, Escher-tiled objects,
fabricated on a Fused-Deposition Modeling (FDM) rapid prototyping machine.
 Escher tiling with 12 lizards (tetrahedral symmetry) Escher tiling with 24 birds (octahedral symmetry) Tiling with 60 butterflies (icosahedral symmetry) 48 starfish on genus-7 surface

# Definitive Project Description (Phase 2):

Functional Optimization for Fair Surface Design (MVS)  by Moreton and Séquin
Try to get clarity on the following issues:
1. What is the underlying representation for Minimum-Variation Surfaces ?
2. What are the various (nested?) loops of the optimization process ?
3. How are G1- and G2-continuity enforced ?