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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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Cs 284
CS 284

Minimum Variation SurfacesCarlo H. Séquin

EECS Computer Science Division

University of California, Berkeley


Smooth surfaces and cad
Smooth 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 nature
Inspiration from Nature

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 s quin 2003
“Volution” Surfaces (Séquin, 2003)

“Volution 0” --- “Volution 5”Minimal surfaces of different genus.


Brakke s surface evolver
Brakke’s Surface Evolver

  • For creating constrained optimized shapes

Start with a crude

polyhedral object

Subdivide triangles

Optimize 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 surfaces
For Closed Manifold Surfaces

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”


Minimum energy surfaces mes
Minimum Energy Surfaces (MES)

  • Lawson surfaces of absolute minimal energy:

12littlelegs

Genus 3

Genus 5

Genus 11

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


Other optimization functionals
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
Minimum-Variation Surfaces (MVS)

  • 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(genus 4 surfaces)






Comparison mes mvs
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 1 st implementation
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 …


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