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Triangle Surfaces w ith DiscreEquivalence Classes
Mayank SinghScott Schaefer
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Introduction
Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007]
Pottmann et al. [2008] SchiftneKillian et al. [2008]
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Introduction
Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007]
Pottmann et al. [2008] SchiftneKillian et al. [2008]
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Introduction
Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007]
Pottmann et al. [2008] SchiftneKillian et al. [2008]
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Introduction
Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007]
Pottmann et al. [2008] SchiftneKillian et al. [2008]
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Introduction
Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007]
Pottmann et al. [2008] SchiftneKillian et al. [2008]
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Economy
Paneling Architectural Freeform SurfacesMichael Eigensatz, Martin Kilian, Alexander Schiftner, Niloy J. Mitra,
Helmut Pottmann and Mark Pauly
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Motivation
Beijing Aquatic Center
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Equivalent Se
576 triangles | 6 un
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Patterns 2D
Planar patterns generated by Craig Kaplan [2004]
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Patterns 3D
Quad parameterization of planar patterns [2009]
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Mosaic 2D
Kim & Pellacini [2002]
Elber & Wolberg [
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Mosaic 3D
Lai et al. [2006]
Passo & Walter [20
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Equivalent Set Surface
OptimizeOriginal
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Discrete Equivalence Classes
Clustering
Rigid Transformation
Global Linear Optimization
ModifiedGeometry
Polygon Assignment &Canonical Triangles
Mesh of CanonicalTriangles
Input Shape
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Example
5-Point Ten
1280 tria
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Canonical Triangle
iP
i
indCD
j
(min,
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Triangle Sim ilarity
a1
a3
a2 b1
b2
b3
Transform B
),( BAD
=
=
+=3
1
2
),(,,
||min),(l
lljpermjTIRR
aTRbBADT
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Triangle Sim ilarity
a1
a3
a2 b1
b2
b3
),( BAD
(b1, b2, b3), ((b3, b1, b2), (
(b3, b2, b1), (
(a1, a2, a3)
D(A,B) = minR
TR =I,T, j
|Rbperm( j,l ) +T al |2
l=1
3
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Canonical Triangle
)0,,(
)0,0,(
)0,0,0(
333,
22,
1,
yxC
xC
C
j
j
j
=
=
=
(0,0,0) (x2,0,0)
(x3,y3,0)
)(iindC
i
indCD
j
(min,Nonlinear Minimization
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Canonical Triangle
Rigid Transfor
=
=
3
1,,
|minl
permjTIRR
RCT
iP
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Adaptive K-Means Clustering
Each triangle is representedas a point
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Adaptive K-Means Clustering
Compute center of the clusterusing nonlinear search
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Adaptive K-Means Clustering
Assign the farthest point to anew cluster
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Adaptive K-Means Clustering
Reassign points to availableclusters
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Adaptive K-Means Clustering
Process continues to generatemore clusters
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Adaptive K-Means Clustering
Process continues to generatemore clusters
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Clustering
Polygon Assignment
Gen
Clus
Nonlinear Optimization
Canonical
Polygons
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Clustering
Error
Number of Clusters
2010
5
1
i
indCD
j
min,
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Clustering
=
=
3
1,,
|minl
permjTIRR
RCT
1280 triangles
Rigid Transfor
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Clustering
1280 triangles |
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Varying the Number of Clusters
1 5
10 20
B
O
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Spacing between Triangles
20 clusters
Before GlobalOptimization
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Disconnected Triangles
Poisson Optimization - Yu et al. [2004]
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Global OptimizationPoisson Optimization
Re-Cluster
Re-ComputeCanonical
Triangles
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Global Optimization
)(min bcgP
EEE ++
Gradient Proximity to originshape
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Prox imity and Fairness
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Prox imity and Fairness
GlobalNon-LinearOptimization
d
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Prox imity and Fairness
=
=
3
1,,
|minl
pjTIRR
RCT
Rigid Tran
GlobalNon-LinearOptimization
RotatTriang
1 Cl
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1 - Cluster
Archit576 T
2 Cl t
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2 - Clusters
3 Cl t
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3 - Clusters
4 Cl t
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4 - Clusters
5 Cl t
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5 - Clusters
6 Clusters
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6 - Clusters
Clustering & Global Optimization
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Clustering & Global Optimization
Before Global Optimization
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Before Global Optimization
1 5
2010
After Global Optimization
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After Global Optimization
1 5
2010
Example
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Example
2492 triangles | 64 clusters = 2.56% of total triangles
Roof
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Roof1.722%
Torus Knot
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Torus Knot2.014%
Venus
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Venus6.017%
Bunny
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Bunny2.436%
4-point roof
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4 point roof
5-point roof
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5 point roof0.781%
Comparison
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Comparison
K-set Tilable Surfaces Ours
Non planar Quadrilaterals Planar Triangle
8 permutations for best rigidtransformation 6 permutations for best rigtransformation
Mean S-quad, compute once Non linear search for canon
Global non-linear optimization Global linear optimization
Begin with large # of clusters & merge Begin with small # of clust
Future Work
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Future Work
Detect outliers in clusters
n-gons
Planarity
Modify topology
Symmetry? Maintain streamlines
Non-existent?
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Paneling Arch. Freeform Surfaces
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Paneling Arch. Freeform Surfaces
Use small # of molds, with associated co
Create non-congruent panels from the m Emphasis upon streamlines
Minimize divergence and kink angle
Clustering
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Clustering
17 Clusters before global optimizati
convergence
Adding 1 Cluster incrementallyand running optimization to
convergence
Rotation of Canonical Triangle
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Rotation of Canonical Triangle
50% rotation 100% rotation
Comparative Analysis
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p y
Paneling ArchitecturalFreeform Surfaces
K-set Tilab le Surfaces Triangle SuDiscrete Eq
Classes
Use of small # of molds Each mold has anassociated cost Emphasis upon
streamlines Divergence and Kinkangle
Non-planar quads 8 permutations for rigidtransformation Global non-linear
optimization Start with large # ofclusters and merge Mean S-quad, computedonce
Planar Tria 6 permutattransformati Global line
Begin withmore Non linear canonical triupdated for