CS 284
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Transcript of CS 284
CS 284CS 284
Minimum Variation Surfaces
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
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
““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 ?
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
““Volution” Surfaces (SVolution” Surfaces (Sééquin, 2003)quin, 2003)
“Volution 0” --- “Volution 5”
Minimal surfaces of different genus.
Brakke’s Surface EvolverBrakke’s Surface Evolver
For creating constrained optimized shapes
Start with a crudepolyhedral object
Subdivide trianglesOptimize vertices
Repeat theprocess
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 !
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”
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
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 …
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
Comparison: Comparison: MES MES MVS MVS(genus 4 surfaces)(genus 4 surfaces)
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
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 …