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 2006, 7/31/2006

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 2006, 7/31/2006

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 2006, 7/31/2006

4. Overlaps Vertex-face: corners Vertex-edge: wedges Edge-face: triangle
Edge-vertex: wedges
Face-vertex: corner quad
Face-edge: triangle
Siggraph 2006, 7/31/2006

5. Transition functions Edge-face: Affine Translate, rotate, translate
Face-vertex: Projective
Square->quadrilateral
Edge-vertex: Composition
Siggraph 2006, 7/31/2006

6. Transition functions Edge-vertex: Blend transition functions
Siggraph 2006, 7/31/2006

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 2006, 7/31/2006

8. Adding geometry Blend function per chart “Bump” covering chart
Partition of unity by dividing by sum of overlapping
Embed function is a spline
Fit to subdivision surface
1-1 correspondence between manifold and dual mesh
Siggraph 2006, 7/31/2006

9. Plusses Embed functions simple, well-behaved Three-chart overlap
Transition functions (mostly) simple
Locality
Siggraph 2006, 7/31/2006

10. Minuses Blending composition function is ugly Difficult to analyze
Large number of charts
Siggraph 2006, 7/31/2006