Connectivity shapes

We describe a method to visualize the connectivity graph of a mesh using a natural embedding in 3D space. This uses a 3D shape representation that is based solely on mesh connectivity: the connectivity shape. Given a connectivity, we define its natural geometry as a smooth embedding in space with uniform edge lengths and describe efficient techniques to compute it. Our main contribution is to demonstrate that a surprising amount of geometric information is implicit in the connectivity. We also show how to generate connectivity shapes that approximate given 3D shapes. Potential applications of connectivity shapes to modeling and mesh coding are described.

[1]  Gabriel Taubin,et al.  Geometric compression through topological surgery , 1998, TOGS.

[2]  Michael T. Goodrich,et al.  A Multi-dimensional Approach to Force-Directed Layouts of Large Graphs , 2000, Graph Drawing.

[3]  Pascal J. Frey,et al.  About Surface Remeshing , 2000, IMR.

[4]  Martin Isenburg,et al.  Face fixer: compressing polygon meshes with properties , 2000, SIGGRAPH.

[5]  Craig Gotsman,et al.  Spectral compression of mesh geometry , 2000, EuroCG.

[6]  Jarek Rossignac,et al.  Edgebreaker: Connectivity Compression for Triangle Meshes , 1999, IEEE Trans. Vis. Comput. Graph..

[7]  W. T. Tutte.  How to Draw a Graph , 1963 .

[8]  Greg Turk,et al.  Re-tiling polygonal surfaces , 1992, SIGGRAPH.

[9]  Marshall W. Bern,et al.  A new Voronoi-based surface reconstruction algorithm , 1998, SIGGRAPH.

[10]  Craig Gotsman,et al.  Efficient Coding of Non-Triangular Meshes , 2000, EuroCG.

[11]  Michael S. Floater,et al.  Parametrization and smooth approximation of surface triangulations , 1997, Comput. Aided Geom. Des..

[12]  Grace Wahba,et al.  Spline Models for Observational Data , 1990 .

[13]  Craig Gotsman,et al.  Triangle Mesh Compression , 1998, Graphics Interface.

[14]  David Harel,et al.  A Fast Multi-scale Method for Drawing Large Graphs , 2000, Graph Drawing.

[15]  Kenneth M. Hall.  An r-Dimensional Quadratic Placement Algorithm , 1970 .

[16]  Wolfgang Straßer,et al.  Real time compression of triangle mesh connectivity , 1998, SIGGRAPH.

[17]  Andrei Khodakovsky,et al.  Progressive geometry compression , 2000, SIGGRAPH.

[18]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..

[19]  Robert Connelly,et al.  A counterexample to the rigidity conjecture for polyhedra , 1977 .

[20]  David Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

[21]  Michael B. Dillencourt,et al.  Realizability of Delaunay Triangulations , 1990, Inf. Process. Lett..

[22]  Michael T. Goodrich,et al.  A Fast Multi-Dimensional Algorithm for Drawing Large Graphs? , 2000 .

[23]  Craig Gotsman,et al.  Efficient coding of non-triangular mesh connectivity , 2000, Proceedings the Eighth Pacific Conference on Computer Graphics and Applications.