Southern Oregon University, May 2003 Surface Optimization and Aesthetic Engineering Carlo Séquin, University of California, Berkeley

I am a Designer … CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994 RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000

Focus of Talk The role of the computer in: u the creative process, u aesthetic optimization.

Outline u Collaboration with Brent Collins u Parameterized Shape Generation u Realization by Layered Manufacturing u Geometric Sculptures in Snow u Aesthetics of Minimal Surfaces u Sphere Inversion as a Challenge u Search for a “Beauty Functional” u CAD Tools that We Are Lacking

Leonardo -- Special Issue On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art With an Artist’s Statement by Brent Collins George K. Francis with Brent Collins

Brent Collins “Hyperbolic Hexagon II”

Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles (monkey saddle)

Brent Collins: Stacked Saddles

“Hyperbolic Hexagon” by B. Collins u 6 saddles in a ring u 6 holes passing through symmetry plane at ±45º u = “wound up” 6-story Scherk tower u Discussion: What if … l we added more stories ? l or introduced a twist before closing the ring ?

Closing the Loop straight or twisted

Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

“Sculpture Generator I”, GUI

A Simple Scherk-Collins Toroid Parameters: (genome) u branches = 2 u stories = 1 u height = 5.00 u flange = 1.00 u thickness = 0.10 u rim_bulge = 1.00 u warp = u twist = 90 u azimuth = 90 u textr_tiles = 3 u detail = 8

A Scherk Tower (on its side) u branches = 7 u stories = 3 u height = 0.2 u flange = 1.00 u thickness = 0.04 u rim_bulge = 0 u warp = 0 u twist = 0 u azimuth = 0 u textr_tiles = 2 u detail = 6

A Virtual Sculpture (1996)

V-art Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

Minimal Surfaces At all surface points, Minimal Surfaces have equal and opposite principal curvatures. Catenoid

Main Goal in Sculpture Generator 1 Real-time Interactive Speed ! u Can’t afford real surface optimization to obtain true minimal surfaces (too slow); u also, this would be aesthetically too limited.  Make closed-form hyperbolic approximation.

Hyperbolic Cross Sections

Base Geometry: One Scherk Story u Hyperbolic Slices  Triangle Strips u precomputed  then warped into toroid

The Basic Saddle Element with surface normals

Hyperbolic Contour Lines On a straight tower and on a toroidal ring

Part III How to Obtain a Real Sculpture ? u Prepare a set of cross-sectional blue prints at equally spaced height intervals, corresponding to the board thickness that Collins is using for the construction.

Collins’ Fabrication Process Example: “Vox Solis” Layered laminated main shape Wood master pattern for sculpture

Emergence of the “Heptoroid” (1) Assembly of the precut boards

Emergence of the “Heptoroid” (3) Smoothing the whole surface

Slices through “Minimal Trefoil” 50%10%23%30% 45%5%20%27% 35%2%15%25%

SFF (Solid Free-form Fabrication) Monkey- Saddle Cinquefoil

Fused Deposition Modeling (FDM)

Zooming into the FDM Machine

Various “Scherk-Collins” Sculptures

Part IV But what, if we want to make a really large sculpture ?

Breckenridge, 2003 Brent Collins and Carlo Séquin are invited to join the team and to provide a design. Other Team Members: Stan Wagon, Dan Schwalbe, Steve Reinmuth (= Team “Minnesota”)

Stan Wagon, Macalester College, St. Paul, MN u Leader of Team “USA – Minnesota”

Breckenridge, 1999 Helaman Ferguson: “Invisible Handshake”

Breckenridge, 2000 Robert Longhurst: “Rhapsody in White” 2 nd Place

Monkey Saddle Trefoil from Sculpture Generator I

The Poor Man’s Opportunity: Snow-Sculpting! Annual Championships in Breckenridge, CO

“Whirled White Web”

12:40 pm -- 42° F

12:40:01 Photo: StRomain

12:41 pm -- 42° F

The Winners 1 st : Canada – B.C., 2 nd : USA – Minnesota, 3 rd : USA – Breckenridge “… sacred geometry … very intricate … very 21 st century !”

4 pm

Snow Sculpting u More on the construction and drama of our snow sculpture tonight at 7pm. u Also, pictures of some of the other snow sculptures.

Part V DISCUSSION: Aesthetics of Minimal Surfaces

“Whirled White Web” Séquin 2003 Minimal surface spanning three (2,1) torus knots Maquette made with Sculpture Generator I

“Tightest Saddle Trefoil” Séquin 1997 Shape generated with Sculpture Generator 1 Minimal surface spanning one (4,3) torus knots

“Atomic Flower II” by Brent Collins Minimal surface in smooth edge (captured by John Sullivan)

Surface by P. J. Stewart (J. Hrdlicka) Minimal surface in three circles Sculpture constructed by hand

“Volution” Shells (Séquin 2003) Genus 0 and genus 1; generated by Surface Evolver

Aesthetics of True Minimal Surfaces u Large-area minimal surfaces are a challenge for any artist to improve on. u For ribbon-like minimal surfaces, the artist typically prefers a deeper channel:  more drama and more strength.

Part VI SNOWSCULPTING PLANS FOR 2004: u A realistic possibility: a type of “Volution” shell. u A really crazy idea: “Turning a Snowball Inside Out”     Discussion of inadequacy of CAD tools

Sphere Eversion u In 1980, the blind mathematician B. Morin, (born 1931) conceived of a way how a sphere can be turned inside-out: l Surface may pass through itself, l but no ripping, puncturing, creasing allowed, e.g., this is not an acceptable solution: PINCH

Morin Surface u But there are more contorted paths that can achieve the desired goal. u The Morin surface is the half-way point of one such path: John Sullivan: “The Optiverse”

Simplest Model Partial cardboard model based on the simplest polyhedral sphere (= cuboctahedron) eversion.

Gridded Models for Transparency 3D-Print from ZcorpSLIDE virtual model

Shape Adaption for Snow Sculpture Restructured Morin surface to fit block size: (10’ x 10’ x 12’)

Shape Optimization u What is the “fairest” surface with the connectivity of the Morin surface that will fill the given bounding box ? u Minimal surfaces are of no help, since this object clearly must have some positive curvature ! u What other functionals could we use ? Is there a “Beauty Functional” ?

Beauty Functional: Desirable Properties u Smoothness: continuous differentiability. u Fairness: even distribution of curvature u Monotonicity preserving: no unnecessary bulges, ripples. u Invariance under rigid-body transforms, uniform scaling. u Stability: small change in specs  small change in shape. u Consistency: no change if extra point is added on the shape. u Technical relevance: leads to spheres, cylinders, cones, tori.

Various Optimization Functionals u Minimum Length / Area: (rubber bands, soap films)  Polygons; -- Minimal Surfaces.  Minimum Bending Energy: (stiff “Elastica”) ∫  2 ds -- ∫   2 2 dA  Splines; -- Minimum Energy Surfaces.  Minumum Curvature Variation: (no natural model ?) ∫ (d  ds  2 ds -- ∫ (d  1  ds  2 + (d  2  ds  2 dA  Circles; -- Cyclides: Spheres, Cones, Tori …  “Minumum Variation Surfaces” (MVS)

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

Optimization With Constraints u Create the fairest possible surface that fits all the given constraints, which could be: l Position: Give points to be interpolated l Normals: Define tangent planes l Curvature: Define a quadric to be matched u Pictures based on implementation 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 (slow!)

Comparison: MES  MVS (genus 4 surfaces)

Comparison MES  MVS Things get worse for MES as we go to higher genus: Genus-5 MES MVS

Another Problem: Make Surface “Transparent” u Realize surface as a grid. u Draw a mesh of smooth lines onto the surface … u Ideally, these are geodesic lines.

Real Geodesics u Chaotic Path produced by a geodesic line on a surface with concave as well as convex regions.

Geodesic Lines u “Fairest” curve is a “straight” line. u On a surface, these are Geodesic lines: (they bend with the given surface, but make no gratuitous lateral turns). u We can easily draw such a curve from an initial point in a given direction: Step-by-step construction of the next point (or of a short line segment). u But connecting two given points on a given surface by a geodesic is an NP-hard problem.

Another Use for “Geodesics” u Map a complex graph onto a genus-3 surface: u Edges of graph should be nice, smooth curves.

Strut Construction in Snow u Drawing lines is not good enough for snow-sculpture; we need struts of substantial thickness. u As few struts as possible should give a good view of the whole smoothly curved surface. u We will cut windows into a smooth surface, so that a network of struts is left standing. u Surface of struts should follow curvature of surface, and their sides should be normal to the surface. u How do we create a CAD model of this ? -- Some kind of sophisticated CSG operation ? u Moreover, the struts in our model should be adjustable in width and in depth …

Best Modeling Effort as of 5/25/03

Haven’t Found Suitable Tools yet … u We are struggling with subdivision surfaces and with sweeps along spline curves … u We have created our own tools in SLIDE (Scene Language for Interactive Dynamic Environments), a research system built in my group. u SLIDE can create surface-grid representations, but only at the chosen sampling density. We need super-sampling to obtain curved struts.

Conceptual Design (3D Sketching) E.g. creating a new form ( a Moebius bridge )… u CAD Tools are totally inadequate. u Effective design ideation involves more than just the eyes and perhaps a (3D?) stylus. u WANTED: full-hand haptics (palm and fingers), whole body gestures, group interactions, …

The Holy Grail of a CAD System (for abstract, geometric sculpture design) u Combines the best of physical / virtual worlds: u No gravity  no scaffolding needed; u Parts have infinite strength  don’t break; u Parts can be glued together – and taken apart; u Beams may bend like perfect splines (or MVC); u Surfaces may stretch like soap films (or MVS); u Parts may emulate materials properties (sound).

QUESTIONS ? DISCUSSION ?