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Grimm and Hughes

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

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Grimm and Hughes

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  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 www.cs.wustl.edu/~cmg

  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 www.cs.wustl.edu/~cmg

  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 www.cs.wustl.edu/~cmg

  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 www.cs.wustl.edu/~cmg

  5. Transition functions • Edge-face: Affine • Translate, rotate, translate • Face-vertex: Projective • Square->quadrilateral • Edge-vertex: Composition Siggraph 2006, 7/31/2006 www.cs.wustl.edu/~cmg

  6. Transition functions • Edge-vertex: Blend transition functions Siggraph 2006, 7/31/2006 www.cs.wustl.edu/~cmg

  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 www.cs.wustl.edu/~cmg

  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 www.cs.wustl.edu/~cmg

  9. Plusses • Embed functions simple, well-behaved • Three-chart overlap • Transition functions (mostly) simple • Locality Siggraph 2006, 7/31/2006 www.cs.wustl.edu/~cmg

  10. Minuses • Blending composition function is ugly • Difficult to analyze • Large number of charts Siggraph 2006, 7/31/2006 www.cs.wustl.edu/~cmg

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