1. 1 Manifolds from meshes Cindy Grimm and John Hughes, “Modeling Surfaces of Arbitrary Topology using Manifolds”, Siggraph ’95 J. Cotrina Navau and N. Pla Garcia, “Modeling surfaces from meshes of arbitrary topology”,, Computer Aided Geometric Design, 2000 Lexing Ying and Denis Zorin, "A simple manifold-based construction of surfaces of arbitrary smoothness", Siggraph ’04

2. 2 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Overview Goal: Construct a smooth, analytical surface from an input sketch mesh

3. 3 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Overview Steps: Build an abstract manifold using the connectivity of the mesh Disks (charts), overlaps, transition functions Assign geometry to each chart Fit geometry to smooth approximation of sketch mesh (subdivision surface) Blend to produce final surface

4. 4 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Goals A high-order surface construction Important for geometric and numerical computation Desirable features C  or C k smoothness At least 3-flexibility at vertices Closed-form smooth local parameterizations Can handle arbitrary control meshes Good visual quality Easy to implement

5. 5 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Smoothness C k smoothness A standard goal in CAGD important for high- accuracy computation Computing surface properties C 1 : needed for normal C 2 : needed for curvatures, reflection lines; C 3 : needed for curvature variation;

6. 6 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Flexibility Ability to represent local geometry Property of surface construction method (dof) Two-Flexibility: any desired curvature at any point 1-flexible2-flexible

7. 7 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Local parameterization Explicit smooth local parameterization For any point, there is an explicit formula f(x,y) defining the surface in a neighborhood of this point Simplifies many tasks Defining functions on surfaces Integration over surfaces Surface-surface intersections Computing geodesics

8. 8 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Spline-based approach Construct surface patch for each face Patches share boundaries Find smooth local parameterization for every point Difficult to guarantee smoothness for points on patch boundaries

9. 9 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Manifold-based approach Construct overlapping charts covering the mesh Build local geometry approximating the mesh on each chart Find blending function for each chart Get the surface by blending local geometry ……

10. 10 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Visual smoothness Large overlap regions Short blend regions look like discontinuities Chart embed function agreement End conditions Parameterization Close to affine No skew, stretch

11. 11 Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html Three techniques: Shared properties Arbitrary topology, including boundary Number of charts determined by number of elements in the sketch mesh E.g., vertices, edges, faces Default disk shape/size E.g., n-sided unit polygon Transition functions only between adjacent elements E.g., a face and its vertices Blend function covers chart Embed functions Subdivision surfaces are used to specify the desired geometry