Martin Isenburg

1
Connectivity Shapes

• Martin Isenburg
• University of North Carolina
• at Chapel Hill

Craig Gotsman Technion - Israel Instituteof
Technology
Stefan Gumhold University of Tübingen
2
Introduction
3
Overview

• Shape from Connectivity
• Connectivity from Shape
• Hierarchical Methods
• Applications
• Graph Drawing
• Compression
• Connectivity Creatures
• Discussion

4
Shape from Connectivity
5
Shape from Connectivity
6
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 ?
  R3that 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
• xi - xj 1 ? ( i , j ) ?
  E
• The number of unknowns is determined by Eulers
  relation

m n f 2g - 1
8
Spring Energy ES

• Minimize
• ES ?( xi - xj - 1 )2 ? ( i , j )
  ? E

9
Roughness Energy ER

• ER ? L( xi )2

10
Final equation
11
Family of Connectivity Shapes
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Optimal Smoothing ?opt
13
Iterative Solver
14
Modified Spring Energy ES

• ES ?( xi - xj 2 - 1 )2 ? ( i , j
  ) ? E

15
Connectivity from Shape
16
Connectivity from Shape
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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
21
Hierarchical Methods
22
Constructing the Hierarchy
23
Applications
24
Mesh Compression
25
Connectivity Creatures
26
End
27
Bloopers