1. Martin Isenburg University of North Carolina at Chapel Hill Craig Gotsman Technion - Israel Institute of Technology Stefan Gumhold University of Tübingen Connectivity Shapes

2. Introduction

3. Overview Shape from Connectivity Connectivity from Shape Hierarchical Methods Applications –Graph Drawing –Compression –Connectivity Creatures Discussion

4. Shape from Connectivity

6. Connectivity Shape Given a connectivity graph C = ( V, E ) consisting of a list vertices V = ( v 1, v 2,..., v n ) and a set undirected edges E = { e 1, e 2,..., e m } : e j = ( i 1, i 2 ) The connectivity shape CS ( C ) of C is a list of vectors ( x 1, x 2, x 3,..., x n ) : x i  R 3 that satisfy some “natural” property.

7. Some “Natural” Property “all edges have unit length”  Equilibrium state of spring system. The connectivity shape is the solution to a set of m equations of the form || x i - x j || = 1  ( i, j )  E The number of unknowns is determined by Euler’s relation m = n + f + 2g - 1

8. Spring Energy E S Minimize E S =  ( || x i - x j || - 1 ) 2  ( i, j )  E

9. Roughness Energy E R E R =  L( x i ) 2

10. Final equation

11. Family of Connectivity Shapes

12. Optimal Smoothing opt opt = argmax Volume ( CS ( C, ) )  [ 0,1 ]

13. Iterative Solver

14. Modified Spring Energy E’ S E’ S =  ( || x i - x j || 2 - 1 ) 2  ( i, j )  E

15. Connectivity from Shape

17. Meshing / Re-meshing objective: generate a faithful approximation of a given shape, but use only edges of unit length we customized Turk method

18. Smoothing Parameter dev

19. Example Run

20. Hierarchical Methods

22. Constructing the Hierarchy

23. Applications

24. Mesh Compression

25. Connectivity Creatures

26. End

27. Bloopers