CS 284
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Transcript of CS 284
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CS 284CS 284
Minimum Variation Surfaces
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
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Smooth Surfaces and CADSmooth Surfaces and CAD
Smooth surfaces play an important role in engineering.
Some are defined almost entirely by their functions Ships hulls
Airplane wings
Others have a mix of function and aesthetic concerns Car bodies
Flower vases
In some cases, aesthetic concerns dominate Abstract mathematical sculpture
Geometrical models TODAY’S FOCUS
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““Beauty” ? Fairness” ?Beauty” ? Fairness” ?
What is a “ beautiful” or “fair” geometrical surface or line ?
Smoothness geometric continuity, at least G2, better yet G3.
No unnecessary undulations.
Symmetry in constraints are maintained.
Inspiration, … Examples ?
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Inspiration from NatureInspiration 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
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““Volution” Surfaces (SVolution” Surfaces (Sééquin, 2003)quin, 2003)
“Volution 0” --- “Volution 5”
Minimal surfaces of different genus.
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Brakke’s Surface EvolverBrakke’s Surface Evolver
For creating constrained optimized shapes
Start with a crudepolyhedral object
Subdivide trianglesOptimize vertices
Repeat theprocess
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Limitations of “Minimal Surfaces”Limitations of “Minimal Surfaces”
“Minimal Surface” - functional works well forlarge-area, open-edge surfaces.
But what should we do for closed manifolds ?
Spheres, tori, higher genus manifolds … cannot be modeled by minimal surfaces.
We need another functional !
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For Closed Manifold SurfacesFor Closed Manifold Surfaces
Use thin-plate (Bernoulli) “Elastica”
Minimize bending energy:
2 ds 12 + 2
2 dA Splines; Minimum Energy Surfaces.
Closely related to minimal area functional:
(1+ 2)2 = 12 + 2
2 + 212
4H2 = Bending Energy + 2G
Integral over Gauss curvature is constant: 212 dA = 4* (1-genus)
Minimizing “Area” minimizes “Bending Energy”
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Minimum Energy Surfaces (MES)Minimum Energy Surfaces (MES)
Lawson surfaces of absolute minimal energy:
Genus 5 Genus 11
Shapes get worse for MES as we go to higher genus …
Genus 3
12littlelegs
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Other Optimization FunctionalsOther Optimization Functionals
Penalize change in curvature !
Minimize Curvature Variation: (no natural model ?)
Minimum Variation Curves (MVC): (dds2 ds Circles.
Minimum Variation Surfaces (MVS): (d1de12 + (d2de22 dA Cyclides: Spheres, Cones, Various Tori …
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Minimum-Variation Surfaces (MVS)Minimum-Variation Surfaces (MVS)
The most pleasing smooth surfaces…
Constrained only by topology, symmetry, size.
Genus 3 D4h Genus 5 Oh
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Comparison: Comparison: MES MES MVS MVS(genus 4 surfaces)(genus 4 surfaces)
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Comparison MES Comparison MES MVS MVS
Things get worse for MES as we go to higher genus:
Genus-5 MES MVSkeep nice toroidal arms
3 holes pinch off
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MVS: 1MVS: 1stst Implementation Implementation
Thesis work by Henry Moreton in 1993:
Used quintic Hermite splines for curves
Used bi-quintic Bézier patches for surfaces
Global optimization of all DoF’s (many!)
Triply nested optimization loop
Penalty functions forcing G1 and G2 continuity
SLOW ! (hours, days!)
But results look very good …