Transcript Minimum Variation Surfaces

CS 284
Minimum Variation
Surfaces
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
Smooth Surfaces and CAD
Smooth surfaces play an important role in engineering.
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Some are defined almost entirely by their functions
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Ships hulls
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Airplane wings
Others have a mix of function and aesthetic concerns
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Car bodies
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Flower vases
In some cases, aesthetic concerns dominate
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Abstract mathematical sculpture
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Geometrical models
TODAY’S FOCUS
“Beauty” ? Fairness” ?
What is a “ beautiful” or “fair” geometrical
surface or line ?
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Smoothness  geometric continuity,
at least G2, better yet G3.
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No unnecessary undulations.
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Symmetry in constraints are maintained.
 Inspiration,
… Examples ?
Inspiration from Nature
Soap films in wire frames:
 Minimal
area
 Balanced
curvature:
k1 = –k2;
mean curvature = 0
Natural beauty functional:
 Minimum
Length / Area:
rubber bands,
soap films
 polygons,
minimal surfaces
 ds = min
 dA = min
“Volution” Surfaces (Séquin, 2003)
“Volution 0”
---
“Volution 5”
Minimal surfaces of different genus.
Brakke’s Surface Evolver
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For creating constrained optimized shapes
Start with a crude
polyhedral object
Subdivide triangles
Optimize vertices
Repeat the
process
Limitations of “Minimal Surfaces”
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“Minimal Surface” - functional
works well for
large-area, open-edge surfaces.
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But what should we do for closed manifolds ?
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Spheres, tori, higher genus manifolds …
cannot be modeled by minimal surfaces.
 We need another functional !
For Closed Manifold Surfaces
Use thin-plate (Bernoulli) “Elastica”
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Minimize bending energy:
 k2 ds
 Splines;
 k12 + k22 dA
Minimum Energy Surfaces.
Closely related to minimal area functional:
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(k1+ k2)2 = k12 + k22 + 2k1k2
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4H2 = Bending Energy + 2G
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Integral over Gauss curvature is constant:
 2k1k2 dA = 4p * (1-genus)
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Minimizing “Area”  minimizes “Bending Energy”
Minimum Energy Surfaces (MES)
 Lawson
surfaces of absolute minimal energy:
12
little
legs
Genus 3
Genus 5
Genus 11
Shapes get worse for MES as we go to higher genus …
Other Optimization Functionals
 Penalize
change in curvature !
 Minimize
Curvature Variation:
(no natural model ?)
Minimum Variation Curves (MVC):
 (dk /ds)2 ds
 Circles.
Minimum Variation Surfaces (MVS):
 (dk1/de1)2 + (dk2/de2)2 dA
 Cyclides: Spheres, Cones, Various Tori …
Minimum-Variation Surfaces (MVS)
Genus 3
D4h
Genus 5
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The most pleasing smooth surfaces…
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Constrained only by topology, symmetry, size.
Oh
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Comparison:
MES   MVS
(genus 4 surfaces)
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Comparison MES  MVS
Things get worse for MES as we go to higher genus:
3 holes
Genus-5 MES
pinch off
MVS
keep nice toroidal arms
MVS: 1st Implementation
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Thesis work by Henry Moreton in 1993:
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Used quintic Hermite splines for curves
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Used bi-quintic Bézier patches for surfaces
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Global optimization of all DoF’s (many!)
Triply nested optimization loop
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Penalty functions forcing G1 and G2 continuity
 SLOW ! (hours, days!)
 But
results look very good …