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. u Some are defined almost entirely by their functions l Ships hulls l Airplane wings u Others have a mix of function and aesthetic concerns l Car bodies l Flower vases u In some cases, aesthetic concerns dominate l Abstract mathematical sculpture l Geometrical models TODAY’S FOCUS

“Beauty” ? Fairness” ? What is a “ beautiful” or “fair” geometrical surface or line ? u Smoothness  geometric continuity, at least G 2, better yet G 3. u No unnecessary undulations. u Symmetry in constraints are maintained. u Inspiration, … Examples ?

Inspiration from Nature Soap films in wire frames: u Minimal area u Balanced curvature: k 1 = –k 2 ; mean curvature = 0 Natural beauty functional: u Minimum Length / 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.

Brakke’s Surface Evolver u For creating constrained optimized shapes Start with a crude polyhedral object Subdivide triangles Optimize vertices Repeat the process

Limitations of “Minimal Surfaces” u “Minimal Surface” - functional works well for large-area, open-edge surfaces. u But what should we do for closed manifolds ? u Spheres, tori, higher genus manifolds … cannot be modeled by minimal surfaces.  We need another functional !

For Closed Manifold Surfaces Use thin-plate (Bernoulli) “Elastica” l Minimize bending energy:   2 ds    2 2 dA  Splines; Minimum Energy Surfaces. Closely related to minimal area functional: (  1 +  2 ) 2 =    1  2 l 4H 2 = Bending Energy + 2G Integral over Gauss curvature is constant:  2  1  2 dA = 4  * (1-genus) l Minimizing “Area”  minimizes “Bending Energy”

Minimum Energy Surfaces (MES) u Lawson surfaces of absolute minimal energy: Genus 5Genus 11 Shapes get worse for MES as we go to higher genus … Genus 3 12 little legs

Other Optimization Functionals u Penalize change in curvature !  Minimize Curvature Variation: (no natural model ?) Minimum Variation Curves (MVC):  (d  ds  2 ds  Circles. Minimum Variation Surfaces (MVS):  (d  1  de 1  2 + (d  2  de 2  2 dA  Cyclides: Spheres, Cones, Various Tori …

Minimum-Variation Surfaces (MVS) u The most pleasing smooth surfaces… u Constrained only by topology, symmetry, size. Genus 3 D 4h Genus 5 OhOh

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

Comparison MES  MVS Things get worse for MES as we go to higher genus: Genus-5 MES MVS keep nice toroidal arms 3 holes pinch off

MVS: 1 st Implementation u Thesis work by Henry Moreton in 1993: l Used quintic Hermite splines for curves l Used bi-quintic Bézier patches for surfaces l Global optimization of all DoF’s (many!) u Triply nested optimization loop l Penalty functions forcing G 1 and G 2 continuity u  SLOW ! (hours, days!) u But results look very good …