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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. Some are defined almost entirely by their functions Ships hulls Airplane wings

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Minimum Variation SurfacesCarlo H. Séquin

EECS Computer Science Division

University of California, Berkeley

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

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 ?

Soap films in wire frames:

• Minimal area

• Balanced curvature: k1 = –k2; mean curvature = 0

Natural beauty functional:

• MinimumLength / 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.

• For creating constrained optimized shapes

polyhedral object

Subdivide triangles

Optimize vertices

Repeat theprocess

• “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 !

Use thin-plate (Bernoulli) “Elastica”

• Minimize bending energy:

•  k2 dsk12 + k22 dA Splines; Minimum Energy Surfaces.

Closely related to minimal area functional:

• (k1+ k2)2 = k12 + k22+ 2k1k2

• 4H2= Bending Energy + 2G

• Integral over Gauss curvature is constant:2k1k2 dA = 4p * (1-genus)

• Minimizing “Area” minimizes “Bending Energy”

• Lawson surfaces of absolute minimal energy:

12littlelegs

Genus 3

Genus 5

Genus 11

Shapes get worse for MES as we go to higher genus …

• 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 …

• The most pleasing smooth surfaces…

• Constrained only by topology, symmetry, size.

D4h

Oh

Genus 3

Genus 5

Comparison: MES   MVS(genus 4 surfaces)





Comparison MES  MVS

Things get worse for MES as we go to higher genus:

pinch off

3 holes

Genus-5 MES

MVSkeep nice toroidal arms

MVS: 1st 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 …