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 …