Connectivity Shapes
Martin Isenburg
Stefan Gumhold
Craig Gotsman
University of North Carolina
at Chapel Hill
University of Tübingen
Technion - Israel Institute
of Technology
Introduction
Overview
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•
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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 a
list 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
 ( i , j ) E
The number of unknowns is determined
by Euler’s relation
m = n + f + 2g - 1
|| xi - xj || = 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