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CS 284

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

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  1. CS 284 Minimum Variation SurfacesCarlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. 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

  3. “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 ?

  4. 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

  5. “Volution” Surfaces (Séquin, 2003) “Volution 0” --- “Volution 5”Minimal surfaces of different genus.

  6. Brakke’s Surface Evolver • For creating constrained optimized shapes Start with a crude polyhedral object Subdivide triangles Optimize vertices Repeat theprocess

  7. 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 !

  8. 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”

  9. 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 …

  10. 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 …

  11. Minimum-Variation Surfaces (MVS) • The most pleasing smooth surfaces… • Constrained only by topology, symmetry, size. D4h Oh Genus 3 Genus 5

  12. Comparison: MES   MVS(genus 4 surfaces)  

  13. 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

  14. 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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