1. Grimm and Hughes Input: arbitrary mesh
Subdivide once (Catmull-Clark) and take dual
Mesh with vertices of valence 4
Charts
One for each vertex, edge, face
Overlaps
Adjacent elements
Eg., vertex with 4 faces, 4 edges
Transition functions
Affine (rotate, translate) or projective where possible
Blend where not
Siggraph 2005, 8/1/2005

2. Motivation Maximize overlap Three chart blend better than two
Co-cycle condition made > 3 hard
Affine transformations
(we got close)
Generalize spline construction process
Blend functions, not points
Siggraph 2005, 8/1/2005

3. Charts Vertex: Square Always valence 4 Edge: Diamond
Diamond shape determined by number of sides of adjacent faces
Face: N-sided unit polygon
Shrunk slightly
Siggraph 2005, 8/1/2005

4. Overlaps Vertex-face: corners Vertex-edge: wedges Edge-face: triangle
Edge-vertex: wedges
Face-vertex: corner quad
Face-edge: triangle
Siggraph 2005, 8/1/2005

5. Transition functions Edge-face: Affine Translate, rotate, translate
Face-vertex: Projective
Square->quadrilateral
Edge-vertex: Composition
Siggraph 2005, 8/1/2005

6. ? Transition functions Edge-vertex: F1 Vertex F2
Siggraph 2005, 8/1/2005

7. Transition functions C¥ continuous everywhere except blend area
Ck in blend area (determined by blend function)
At most three charts overlap anywhere
Reflexive: Use identity function
Symmetric: E-F, V-F both invertible
Co-cycle condition satisfied by blend function
Siggraph 2005, 8/1/2005

8. Abstract manifold
Barycentric coords
“Glue” points that are related through transition functions
Each point appears in (at most) one each of a vertex, edge, and face chart f f v e v e
Siggraph 2005, 8/1/2005

9. Adding geometry Blend and embed functions per chart
Fit to subdivision surface
1-1 correspondence between manifold and dual mesh
Siggraph 2005, 8/1/2005

10. Blend functions Vertex: Spline surface basis function
Edge/face: Spline basis function “spun” in a circle
Inscribe
Partition of unity formed by dividing by sum of non-zero basis functions
Charts overlap sufficiently to guarantee sum non-zero
Siggraph 2005, 8/1/2005

11. Embed functions Spline patches Fit to subdivision surface
One-to-one correspondence between dual mesh and charts
Implies correspondence between dual mesh and points in the chart
Siggraph 2005, 8/1/2005

12. Plusses Embed functions simple, well-behaved Three-chart overlap
Transition functions (mostly) simple
Locality
Siggraph 2005, 8/1/2005

13. Minuses Blending composition function is ugly Difficult to analyze
Large number of charts
Siggraph 2005, 8/1/2005