Martin Isenburg

University of North Carolina

at Chapel Hill

Craig Gotsman

Technion - Israel Instituteof 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

Shape from Connectivity

Connectivity Shape

Given a connectivity graph C = ( V, E )consisting of a list vertices

V = ( v1 , v2 , ... , vn )and a set undirected edges

E = { e1 , e2 , ... , em } : ej = ( i1 , i2 )

The connectivity shape CS ( C ) of C is alist of vectors ( x1 , x2 , x3 , ... , xn ) : xi R3

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

|| xi - xj || = 1 ( i , j ) E

The number of unknowns is determined by Euler’s relation m = n + f + 2g - 1

Spring Energy ES

Minimize

ES = ( || xi - xj || - 1 )2 ( i , j ) E

Roughness Energy ER

ER = L( xi )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 = ( || xi - xj ||2 - 1 )2 ( i , j ) E

Connectivity from Shape

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

Hierarchical Methods

Constructing the Hierarchy

Applications

Mesh Compression

Connectivity Creatures

End

Bloopers