Solid Modeling Symposium, Seattle 2003Solid Modeling Symposium, Seattle 2003

Aesthetic Engineering

Carlo H. Séquin

EECS Computer Science Division

University of California, Berkeley

I Am Not an ArtistI Am Not an Artist

I am a Designer, Engineer …I am a Designer, Engineer …

CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994

RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000

““Artistic Geometry”Artistic Geometry”

The role of the computer in:

the creative process,

aesthetic optimization.

Interactivity !

What Drives My Research ?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 TasksOutline: 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 ProblemsOutline: 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 #1TASK GROUP #1Two Graph-Mapping ProblemsTwo Graph-Mapping Problems

(courtesy of Prof. J(courtesy of Prof. Jüürgen Bokowski)rgen 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 KGraph K1212

Mapping Graph KMapping Graph K1212 onto a Surface onto a Surface

(i.e., an orientable two-manifold)(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 ModelBokowski’s Goose-Neck Model

Bokowski’s Bokowski’s ( Partial ) ( Partial )

Virtual Model Virtual Model on a on a

Genus 6 Genus 6 SurfaceSurface

My ModelMy Model

Find highest-symmetry genus-6 surface,

with “convenient” handles to route edges.

My Model (cont.)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)My Model (3)

Now need to place 66 edges:

Use trial and error.

Need a 3D model !

No nice CAD model yet.

A 2A 2ndnd Problem : Dyck’s Graph 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 GraphAnother View of Dyck’s Graph

Difficult to connect up matching nodes !

Folding It into a Self-intersecting PolyhedronFolding It into a Self-intersecting Polyhedron

Towards a 3D ModelTowards a 3D Model Find highest-symmetry genus-3 surface:

Klein Surface (tetrahedral frame).

Find Locations for VerticesFind 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 themissing edges do not break the symmetry:

Inside and outside on each tetra-arm.

Do not connect the vertices that lie on thesame symmetry axis(same color)(or this one).

A First Physical ModelA 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 ?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 SurfacesConstruction of Fair Surfaces

Input: Genus, symmetry class, size;

Output: “Fairest” surface possible: Highest symmetry: G3 Tetrahedral

Smooth: Gn continuous (n2)

Simple: No unnecessary undulations

Good parametrization: (for texturing)

Representation: Efficient, for visualization, RP

Use some optimization process…

Is there a “Beauty Functional” ?

Various Optimization FunctionalsVarious Optimization Functionals

Minimum Length / Area: (rubber bands, soap films) Polygons; -- Minimal Surfaces.

Minimum Bending Energy: (thin plates, “Elastica”) 2 ds -- 1

2 + 22 dA

Splines; -- Minimum Energy Surfaces.

Minumum Curvature Variation: (no natural model ?) (dds2 ds -- (d1de12 + (d2de22 dA Circles; -- Cyclides: Spheres, Cones, Tori …

Minumum Variation Curves / Surfaces (MVC, MVS)

Minimum-Variation SurfacesMinimum-Variation Surfaces

The most pleasing smooth surfaces…

Constrained only by topology, symmetry, size.

Genus 3 D4h Genus 5 Oh

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

Comparison MES Comparison MES MVS MVS

Things get worse for MES as we go to higher genus:

Genus-5 MESMVS

3 holes pinch off

11stst Implementation: Henry Moreton 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?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 …#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 LinesGeodesic 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).

PolyhedralApproximation

Real GeodesicsReal Geodesics

Chaotic Pathproduced by a geodesic lineon a surfacewith saddlesas well as convex regions.

Geodesic Line Between 2 PointsGeodesic Line Between 2 Points

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

T

S

Send geodesic path from S towards T

Vary starting direction; do binary search for hit.

Try: Target-ShootingTry: Target-Shooting

TVSS

T

V

Problem:Where Gauss curvature > 0 (bumps, bowls) two possible paths focussing effect.

Target-Shooting Problem (2)Target-Shooting Problem (2)

Where Gauss curvature < 0 (saddle regions)

no (stable) path defocussing effect.

V

T1

T2

TV

T1

T2S

T1, T2 can only be reached by going through V !

S

Polyhedral Angle AmbiguityPolyhedral 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”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)LVC Curves (instead of MVC)

Curves with linearly varying curvaturehave two degrees of freedom: kA kB,

Allows to set two additional parameters,i.e., the start / ending tangent directions.

A

B

CURVATURE

kA

kB

ARC-LENGTH

The Complete “Shooting Game”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 FindingMore Promising Approach to Findinga “Geodesic” LVC Connectiona “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 pointswhile approaching the LVC curve with the desired start / end tangents.

Path-Optimization towards LVCPath-Optimization towards LVC

Locally move locations of edge crossingsso as to even out variation of curvature:

T

CS

C

As path moves across a vertex, re-analyze the gradient on the new edges, and exploit angle ambiguity.

V

TASK GROUP # 2TASK GROUP # 2

Making RP Models of Math SurfacesMaking RP Models of Math Surfaces

Klein Bottles

Boy’s Surface

Morin Surface

…

Intriguing, self-intersecting in 3D

““Skeleton of Klein Bottle”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 BottleTriply Twisted Figure-8 Klein Bottle

Strut intersections can be avoided by design because of simplicity of intersection line and regularity of strut crossings.

SEQUIN 2000

Avoiding Self-intersectionsAvoiding Self-intersections

Rectangular surface domain of Klein bottle.

Arrange strut patternas shown on the left.

After the figure-8 fold, struts pass smoothly through one another.

A Look into the FDM MachineA Look into the FDM Machine

Triply Twisted Figure-8 Klein BottleTriply Twisted Figure-8 Klein Bottle

As it comes out of the FDM machine

The Finished Klein Bottle The Finished Klein Bottle (supports removed)(supports removed)

The Projective PlaneThe Projective Plane

C

PROJECTIVE PLANE

-- Walk off to infinity -- and beyond … come back upside-down from opposite direction.

Projective Plane is single-sided; has no edges.

Model of Boy SurfaceModel of Boy Surface

Computer graphics by John Sullivan (1998)

Double Covering of Boy SurfaceDouble Covering of Boy Surface

Wire model byCharles Pugh( ~ 1980 )

Decorated by C. H. Séquin:

“Equator”

3 “Meridians,” 120º apart

Can We Avoid Strut IntersectionsCan We Avoid Strut Intersectionsfor Boy’s Surface ?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 EvolverTessellation from Surface Evolver

Triangulation from optimal polyhedron.

Mesh dualization.

Strut thickening.

FDM fabrication.

Quad facet !

Intersecting struts.

Paper Model with Regular TilesPaper 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 TriconnectorA Modular Triconnector

Prototype made in the FDM machine

Assembly of the “Tiled” Boy SurfaceAssembly of the “Tiled” Boy Surface

KIHA LEE

Boy Surface in OberwolfachBoy Surface in Oberwolfach

Sculpture constructed by Mercedes Benz

Photo courtesy John Sullivan

TASK GROUP #3TASK GROUP #3

Combining Math Model MakingCombining Math Model Makingwith some artistic ambitionswith some artistic ambitions

This needs some background …

Brent CollinsBrent Collins

“Hyperbolic Hexagon II”

Brent Collins: Stacked SaddlesBrent Collins: Stacked Saddles

Scherk’s 2nd Minimal SurfaceScherk’s 2nd Minimal Surface

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)

““Hyperbolic Hexagon” by B. CollinsHyperbolic 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 LoopClosing the Loop

straight

or

twisted

Brent Collins’ Prototyping ProcessBrent Collins’ Prototyping Process

Armature for the "Hyperbolic Heptagon"

Mockup for the "Saddle Trefoil"

Time-consuming ! (1-3 weeks)

““Sculpture Generator I”, GUI Sculpture Generator I”, GUI

V-artV-art

VirtualGlassScherkTowerwithMonkeySaddles

(Radiance 40 hours)

Jane Yen

Collins’ Fabrication ProcessCollins’ Fabrication Process

Example: “Vox Solis”

Layered laminated main shapeWood master pattern

for sculpture

Slices through “Minimal Trefoil”Slices through “Minimal Trefoil”

50% 10%23%30%

45% 5%20%27%

35% 2%15%25%

One thick slicethru 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, 3/4of one board.

Profiled Slice through “Heptoroid”Profiled Slice through “Heptoroid”

Emergence of the “Heptoroid” (1)Emergence of the “Heptoroid” (1)

Assembly of the precut boards

Emergence of the “Heptoroid” (2)Emergence of the “Heptoroid” (2)

Forming a continuous smooth edge

Emergence of the “Heptoroid” (3)Emergence of the “Heptoroid” (3)

Smoothing the whole surface

The Finished The Finished “Heptoroid”“Heptoroid”

at Fermi Lab Art Gallery (1998).

Various “Scherk-Collins” SculpturesVarious “Scherk-Collins” Sculptures

Hyper-Sculpture: “Family of 12 Trefoils”Hyper-Sculpture: “Family of 12 Trefoils”

W=2

W=1

B=1 B=2 B=3 B=4

““Cohesion”Cohesion”

SIGGRAPH’2004 Art Gallery

Stan Wagon, Stan Wagon, Macalester College, St. Paul, MNMacalester College, St. Paul, MN

Leader of Team “USA – Minnesota”

Snow-Sculpting, Breckenridge, 2003Snow-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”)

Breckenridge, CO, 1999Breckenridge, CO, 1999

Helaman Ferguson: “Invisible Handshake”

Breckenridge, Breckenridge, 20002000

Robert Longhurst:

“Rhapsody in White”

2nd Place

Monkey Saddle TrefoilMonkey Saddle Trefoil

from Sculpture Generator I

Annual Championships in Breckenridge, COAnnual Championships in Breckenridge, CO

Day 1: The “Monolith”Day 1: The “Monolith”

Cut away prisms …

End of Day 2End of Day 2

The Torus

Day 3, 4: Carving the Flanges, Holes Day 3, 4: Carving the Flanges, Holes

Day 5, am: Surface RefinementDay 5, am: Surface Refinement

““Whirled White Web”Whirled White Web”

12:40 pm -- 4212:40 pm -- 42° F° F

12:41 pm -- 4212:41 pm -- 42° F° F

12:40:0112:40:01

Photo:StRomain

3 pm: “WWW” Wins Silver Medal3 pm: “WWW” Wins Silver Medal

Snow-Sculpting Plans for 2004Snow-Sculpting Plans for 2004

“Turning a Snowball Inside Out”

Design is due July 1, 2003

Again, I am having some problemsmaking a good CAD model.

Sphere EversionSphere 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 SurfaceMorin 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 ModelSimplest Model

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

Gridded Models for TransparencyGridded Models for Transparency

3D-Print from Zcorp SLIDE virtual model

Shape Adaption for Snow SculptureShape Adaption for Snow Sculpture

Restructured Morin surface to fit block size: (10’ x 10’ x 12’)

Make Surface “Transparent”Make Surface “Transparent”

Realize surface as a grid.

Draw a mesh of smooth lines onto the surface …

Ideally, these areLVC lines.

Best Modeling Effort as of 5/25/03Best Modeling Effort as of 5/25/03

Used Sweep-Morph for best controlUsed Sweep-Morph for best controlof placing parameter lines.of placing parameter lines.

Developed a special offset-surface generator Developed a special offset-surface generator that cuts “windows” into all the facets,that cuts “windows” into all the facets,so that only a grid structure remains.so that only a grid structure remains.

Latest FDM Model 6/1/03Latest 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 ToolsCAD 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 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 My Dream of a CAD System of a CAD System (for abstract, geometric sculpture design)(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.

ConclusionsConclusions

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 ?QUESTIONS ?

DISCUSSION ?DISCUSSION ?