Solid Modeling Symposium, Seattle 2003 Florida 1999 Solid Modeling Symposium, Seattle 2003 Aesthetic Engineering Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

I Am Not an Artist

I am a Designer, Engineer … Florida 1999 I am a Designer, Engineer … CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994 … and I like to build things! RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000

“Artistic Geometry” Interactivity ! The role of the computer in: Florida 1999 “Artistic Geometry” The role of the computer in: the creative process, aesthetic optimization. Interactivity ! NOT: “Geometric Art” -- Often my primary purpose is not to make “Art” Some of my efforts are devoted to making Mathematical Visualization Models.

What Drives My Research ? Whatever I need most urgently to get a real job done. Most of my jobs involve building things -- not just pretty pictures on a CRT. Today: Report on some ongoing activities: -- motivation and progress so far. Thanks to: Ling Xiao, Ryo Takahashi, Alex Kozlowski.

Outline: Three Defining Tasks #1: Mapping graphs onto surfaces of suitable genus with a high degree of symmetry. #2: Making models of self-intersecting surfaces such as Klein-bottles, Boy Surface, Morin Surface … #3: Coming up with an interesting and doable design for a snow-sculpture for January 2004.

Outline: Some Common Problems #A: “Which is the fairest (surface) of them all ?” #B: Drawing geodesic lines (or curves with linearly varying curvature) between two points on a surface. #C: Making gridded surface representations (different needs for different applications).

TASK GROUP #1 Two Graph-Mapping Problems (courtesy of Prof TASK GROUP #1 Two Graph-Mapping Problems (courtesy of Prof. Jürgen Bokowski) Given some abstract graph: “K12” = complete graph with 12 vertices, “Dyck Graph” (12vertices, but only 48 edges) Embed each of these graphs crossing-free in a surface with lowest possible genus, so that an orientable matroid results, maintaining as much symmetry as possible.

Graph K12

Mapping Graph K12 onto a Surface (i.e., an orientable two-manifold) Draw complete graph with 12 nodes Has 66 edges Orientable matroid has 44 triangular facets Euler: E – V – F + 2 = 2*Genus 66 – 12 – 44 + 2 = 12  Genus = 6  Now make a (nice) model of that !

Bokowski’s Goose-Neck Model Florida 1999 Bokowski’s Goose-Neck Model Can’t see the triangles – unless you have a vivid imagination.

Bokowski’s ( Partial ) Virtual Model on a Genus 6 Surface

My Model Find highest-symmetry genus-6 surface, with “convenient” handles to route edges.

My Model (cont.) Find suitable locations for twelve vertices: Maintain symmetry! Put nodes at saddle points, because of 11 outgoing edges, and 11 triangles between them.

My Model (3) Now need to place 66 edges: Use trial and error. Need a 3D model ! No nice CAD model yet.

A 2nd Problem : Dyck’s Graph 12 vertices, but only 48 edges. E – V – F + 2 = 2*Genus 48 – 12 – 32 + 2 = 6  Genus = 3

Another View of Dyck’s Graph Difficult to connect up matching nodes !

Folding It into a Self-intersecting Polyhedron

Towards a 3D Model Find highest-symmetry genus-3 surface:  Klein Surface (tetrahedral frame).

Find Locations for Vertices Actually harder than in previous example, not all vertices connected to one another. (Every vertex has 3 that it is not connected to.) Place them so that the missing edges do not break the symmetry:  Inside and outside on each tetra-arm. Do not connect the vertices that lie on the same symmetry axis (same color) (or this one).

A First Physical Model Edges of graph should be nice, smooth curves. Quickest way to get a model: Painting a physical object.

What Are the CAD Tasks Here ? 1) Make a fair surface of given genus. 2) Symmetrically place vertices on it. 3) Draw “geodesic” lines between points. 4) Color all regions based on symmetry.  Let’s address tasks 1) and 3)

Construction of Fair Surfaces Input: Genus, symmetry class, size; Output: “Fairest” surface possible: Highest symmetry: G3  Tetrahedral Smooth: Gn continuous (n2) Simple: No unnecessary undulations Good parametrization: (for texturing) Representation: Efficient, for visualization, RP  Use some optimization process…  Is there a “Beauty Functional” ?

Various Optimization Functionals Minimum Length / Area: (rubber bands, soap films)  Polygons; -- Minimal Surfaces. Minimum Bending Energy: (thin plates, “Elastica”)  k2 ds --  k12 + k22 dA  Splines; -- Minimum Energy Surfaces. Minumum Curvature Variation: (no natural model ?)  (dk / ds)2 ds --  (dk1/de1)2 + (dk2/de2)2 dA  Circles; -- Cyclides: Spheres, Cones, Tori …  Minumum Variation Curves / Surfaces (MVC, MVS)

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

Comparison: MES  MVS (genus 4 surfaces)

Things get worse for MES as we go to higher genus: Comparison MES  MVS Things get worse for MES as we go to higher genus: pinch off 3 holes Genus-5 MES MVS

1st Implementation: Henry Moreton 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 …

What Can Be Improved? Continuity by construction: E.g., Subdivision surfaces Avoids need for penalty functions Improves convergence speed (>100x) Hierarchical approach: Find rough shape first, then refine Further improves speed (>10x) Computers are 100x faster than 1993:  >105  Days become seconds !

#B: Drawing onto that Surface … MVS gives us a good shape for the surface. Now we want to draw nice, smooth curves: They look like geodesics …

Geodesic Lines “Fairest” curve is a “straight” line. On a surface, these are geodesic lines: They bend with the given surface, but make no gratuitous lateral turns. We can easily draw such a curve from an initial point in a given direction: Step-by-step construction of the next point (one line segment per polyhedron facet). Polyhedral Approximation

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

Geodesic Line Between 2 Points Connecting two given points with the shortest geodesic on a high-genus surface is an NP-hard problem.

Try: Target-Shooting Send geodesic path from S towards T Vary starting direction; do binary search for hit. T V S Problem: Where Gauss curvature > 0 (bumps, bowls)  two possible paths  focussing effect.

Target-Shooting Problem (2) Where Gauss curvature < 0 (saddle regions)  no (stable) path  defocussing effect. T1 T2 S V T2 S V T T1 T1, T2 can only be reached by going through V !

Polyhedral Angle Ambiguity At non-planar vertices in a polyhedral surface there is an angle deficit (G>0) or excess (G<0). Whenever a path “hits” a vertex, we can choose within this angle, how the path should continue. If, in our binary search for a target hit, the path steps across a vertex, we can lock the path to that vertex, and start a new “shooting game” from there.

“Pseudo Geodesics” Need more control than geodesics can offer. Want to space the departing curves from a vertex more evenly, avoid very acute angles. Need control over starting and ending tangent directions (like Hermite spline).

LVC Curves (instead of MVC) Curves with linearly varying curvature have two degrees of freedom: kA kB, Allows to set two additional parameters, i.e., the start / ending tangent directions. CURVATURE kB ARC-LENGTH kA B A

The Complete “Shooting Game” Alternate shooting from both ends, gradually adjusting the two end-curvature parameters until the two points are connected and the two specified tangent directions are met. Need to worry about angle ambiguity, whenever the path correction “jumps” over a vertex of the polyhedron. Gets too complicated; instabilities … ==> NOT RECOMMENDED !

More Promising Approach to Finding a “Geodesic” LVC Connection Assume, you already have some path that connects the two points with the desired route on the surface (going around the right handles). Move all the facet edge crossing points so as to even out the curvature differences between neighboring path sample points while approaching the LVC curve with the desired start / end tangents.

Path-Optimization towards LVC Locally move locations of edge crossings so as to even out variation of curvature: S C V C T As path moves across a vertex, re-analyze the gradient on the new edges, and exploit angle ambiguity.

TASK GROUP # 2 Making RP Models of Math Surfaces Klein Bottles Boy’s Surface Morin Surface … Intriguing, self-intersecting in 3D

“Skeleton of Klein Bottle” “Transparency” in the dark old ages when I could only make B&W prints: Take a grid-approach to depicting transparent surfaces. Need to find a good parametrization, which defines nicely placed grid lines. Ideally, avoid intersections of struts (not achieved in this figure). SEQUIN, 1981

Triply Twisted Figure-8 Klein Bottle SEQUIN 2000 Strut intersections can be avoided by design because of simplicity of intersection line and regularity of strut crossings.

Avoiding Self-intersections Rectangular surface domain of Klein bottle. Arrange strut pattern as shown on the left. After the figure-8 fold, struts pass smoothly through one another.

A Look into the FDM Machine

Triply Twisted Figure-8 Klein Bottle As it comes out of the FDM machine

The Finished Klein Bottle (supports removed)

The Projective Plane Projective Plane is single-sided; has no edges. -- Walk off to infinity -- and beyond … come back upside-down from opposite direction. Projective Plane is single-sided; has no edges.

Model of Boy Surface Computer graphics by John Sullivan (1998)

Double Covering of Boy Surface Wire model by Charles Pugh ( ~ 1980 ) Decorated by C. H. Séquin: “Equator” 3 “Meridians,” 120º apart

Can We Avoid Strut Intersections for Boy’s Surface ? This is much harder: More difficult to find a nice, regularly gridded parametrization, Intersection lines are more complicated, Harder to predict where parameter lines will cross over.

Tessellation from Surface Evolver Triangulation from optimal polyhedron. Mesh dualization. Strut thickening. FDM fabrication. Quad facet ! Intersecting struts.

Paper Model with Regular Tiles Only vertices of valence 3. Only meshes with 5, 6, or 7 sides. Struts pass through holes. --> Permits the use of a modular component...

A Modular Triconnector Prototype made in the FDM machine

Assembly of the “Tiled” Boy Surface KIHA LEE

Boy Surface in Oberwolfach Sculpture constructed by Mercedes Benz Photo courtesy John Sullivan

TASK GROUP #3 Combining Math Model Making with some artistic ambitions This needs some background …

Brent Collins “Hyperbolic Hexagon II” Florida 1999 Brent Collins For whom I designed certain shapes on the computer which he then built in wood. “Hyperbolic Hexagon II”

Brent Collins: Stacked Saddles

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

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

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

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

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

Profiled Slice through “Heptoroid” Florida 1999 Profiled Slice through “Heptoroid” One thick slice thru sculpture, from which Brent can cut boards and assemble a rough shape. Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board. From these Collins will precut boards then assemble the complete shape and fine tune and polish it.

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

Emergence of the “Heptoroid” (2) Forming a continuous smooth edge

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

The Finished “Heptoroid” at Fermi Lab Art Gallery (1998).

Various “Scherk-Collins” Sculptures

Hyper-Sculpture: “Family of 12 Trefoils” W=2 W=1 B=1 B=2 B=3 B=4

“Cohesion” SIGGRAPH’2004 Art Gallery

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

Snow-Sculpting, 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”)

Helaman Ferguson: “Invisible Handshake” Breckenridge, CO, 1999 Helaman Ferguson: “Invisible Handshake”

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

from Sculpture Generator I Monkey Saddle Trefoil from Sculpture Generator I

Annual Championships in Breckenridge, CO

Day 1: The “Monolith” Cut away prisms …

End of Day 2 The Torus

Day 3, 4: Carving the Flanges, Holes

Day 5, am: Surface Refinement

“Whirled White Web”

12:40 pm -- 42° F

12:41 pm -- 42° F

12:40:01 Photo: StRomain

3 pm: “WWW” Wins Silver Medal

Snow-Sculpting Plans for 2004 “Turning a Snowball Inside Out” Design is due July 1, 2003 Again, I am having some problems making a good CAD model.

Sphere Eversion ~ 1960, the blind mathematician B. Morin, (born 1931) conceived of a way how a sphere can be turned inside-out: Surface may pass through itself, but no ripping, puncturing, creasing allowed, e.g., this is not an acceptable solution: PINCH

Morin Surface But there are more contorted paths that can achieve the desired goal. 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 Zcorp SLIDE virtual model

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

Make Surface “Transparent” Realize surface as a grid. Draw a mesh of smooth lines onto the surface … Ideally, these are LVC lines.

Best Modeling Effort as of 5/25/03 Used Sweep-Morph for best control of placing parameter lines. Developed a special offset-surface generator that cuts “windows” into all the facets, so that only a grid structure remains.

Latest FDM Model 6/1/03 Work to Be Done: Need a perfect CAD model for bronze cast. Struts should be curved and follow surface. Should be of uniform thickness. Could involve challenging CSG operation. Plan: Build into offset-surface generator.

CAD and Modeling Tools State of the art is lacking … Fairly generic utilities are missing: Surface optimization, Geodesic lines, Gridded surface representations. We are building our own procedural extensions to fill this void.

Tools for Early Conceptual Design For creating new forms, e.g. a “Moebius bridge” 3D “Sketching” Tools are totally inadequate. I typically find myself using cardboard, wires, scotch-tape, styrofoam, clay, wiremesh … Effective design ideation involves more than just the eyes and perhaps a (3D?) stylus.

My Dream of a CAD System (for abstract, geometric sculpture design) Combines the best of virtual / physical worlds: No gravity  no scaffolding needed, Parts have infinite strength  don’t break, Parts can be glued together – and taken apart. Has built-in optimization functionality: Beams may bend like steel wires (or MVC), Surfaces may stretch like soap films (or MVS), Geodesic threads on surfaces. Provides a “hands-on” feel during modeling process. As much co-located haptic feedback as possible.

Conclusions A glimpse of research in progress, what motivates me and my students, and how we tackle some practical problems. This is a solicitation for help with: references to similar work, suggestions of better approaches, or outright collaboration.

QUESTIONS ? DISCUSSION ?