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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

Transition functions Edge-face: Affine Translate, rotate, translate Face-vertex: Projective Square->quadrilateral Edge-vertex: Composition Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

? Transition functions Edge-vertex: F1 Vertex F2 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

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 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

Plusses Embed functions simple, well-behaved Three-chart overlap Transition functions (mostly) simple Locality Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

Minuses Blending composition function is ugly Difficult to analyze Large number of charts Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html