Lecture #15 -- Thu 10/11, 2012.

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Preparation: **Read: **SB'10: An intuitive explanation of third-order surface behavior.**

What is the second fundamental form?

What is the curvature tensor?

How would you measure _

How do you take the _

How do you like to think about the local geometry around a surface point?

Everybody seems to have their subdivision algorithm working nicely:

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).

To model a "spherical" ball we may start with a simple cube; but the "cubist" fish shape will extend beyond a single face and overlap into two adjacent faces.

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! (

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,

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 |

How do the sizes of the triangles in the original (level 0) mesh affect the outcome of the generated texture pattern?

Can you use the "fish" tiles shown above to texture-map your mug? -- What issues might arise?

How would you have to modify the tiles to be able to texture-map an infinite plane?

How to smoothly undo a Klein bottle mouth of the "Inverted Double-Sock" type into a twisted tube with a figure-8 profile?

Some pictures to explain the problem ...

Keep this problem in mind next week when we discuss surface "optimization" (deformation while minimizing some functional).

**Programming: Continue implementing your dyadic triangle subdivision:
1.) Add some rounding/smoothing scheme and demonstrate it on your mug.
Send me pictures by Tuesday, October 16.
2.) Add some "interesting" texture, i.e., the tiles should have some
"directionality" and not fit together with arbitrary rotations or with
every possible pairing of sides. For instance, try to use the square
"fish" tile or the triangular "lizard tile" shown above.
Get this to work by **

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