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

Introduction

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

Shape from Connectivity

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.

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

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

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

Final equation

Family of Connectivity Shapes

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

Iterative Solver

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

Connectivity from Shape

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

Smoothing Parameter dev

Example Run

Hierarchical Methods

Constructing the Hierarchy

Applications

Mesh Compression

Connectivity Creatures

End

Bloopers