Adobe Systems, Strategic Visit, 11/1/06Adobe Systems, Strategic Visit, 11/1/06

Artistic Geometry

Carlo H. Séquin

University of California, Berkeley

A designer for 30 years…A designer for 30 years…

Geometry is my true love !Geometry is my true love !

Optical paths through prism

Integrated circuit layouts

Building floor plans

CAD drawings

Visualizations ...

Fascination with Geometrical Patterns:Fascination with Geometrical Patterns: Escher TilingsEscher Tilings

in the plane on the sphere on the torus

M.C. Escher Jane Yen, 1997 Young Shon, 2002

Can we do this on surfaces of higher genus ?

serious distortions,

not all tiles are the same.Difficulties:

““Infinite” Tiling Infinite” Tiling on the on the PoincarPoincaréé DiskDisk

MappedMappedonto aonto agenus-3genus-3surfacesurface

168 auto-morphisms

Felix Klein (1878)

next:Hurwitz group of order 504

The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)A plane-filling Peano curve

Fall 1983: CS Graduate Course: “Creative Geometric Modeling”

Do This In 3 D !

Metal Sculpture at SIGGRAPH 2006Metal Sculpture at SIGGRAPH 2006

Since 1994: CAD for ArtistsSince 1994: CAD for Artists

How can we use the visualization power offered by computer graphics and by computer-controlled rapid prototyping for the design of geometrical sculptures?

Geometric SculptureGeometric Sculpture

Brent Collins

Art - Math ConnectionArt - Math Connection

ISAMA & Bridges Conferences

MOSAIC 2000 (David Salesin, Univ. of WA)

Many Math Models and SculpturesMany Math Models and Sculptures

CAD tools are still inadequate

for free-form 3D designs !

Ongoing Research InterestOngoing Research Interest

CAD tools for Ideation, Informal Prototyping: Mimick the best of: clay, wire, paper, scotch-tape, styrofoam …

Without the adversity of: messy glue, gravity, strength limits …

Make available pseudo-physical materials that bend as nicely as steel wire, and stretch like a nylon hose, but are strong as titanium, and as transparent as quartz …

Recent Project with Prof. McMains (Recent Project with Prof. McMains (ME)ME)

Develop a haptic, immersive VR work station for telecollaborative design across the internet

See in 3D,Touch, feel,Annotate,Modify,Share, Discuss …

Research with Prof. WrightResearch with Prof. Wright (ME) (ME)

Physical Rapid Prototypes:

For early user testingand hands-on feedbackin application context.

Zcorp 3D-printer FDM

“From Art to Part”

“Contact-Compact”

Current Research (Pushkar Joshi)Current Research (Pushkar Joshi)

Based on optimization of a functionalby using all free DOFs.

Fewer controls to manipulate

Explicit consideration of constraints

Automatic attainment of some quality objectives.

Better Surface Design Tools:

(Future) Surface Optimization Module(Future) Surface Optimization Module Built into future CAD environments!

User specifies all boundary conditions as well as any other constraints.

User selects a suitable “beauty” functional.

CAD system finds best solution in seconds.

““Beauty Functionals”Beauty Functionals”Comparison MES Comparison MES MVS MVS ..

Things get worse for MES as we go to higher genus

Genus-5 MES MVSkeep nice toroidal arms

3 holes pinch off

Pushkar Joshi’s WorkPushkar Joshi’s Work

Build a robust evaluation environmentfor curvature-based surface functionals.

Create a catalog of minimizing shapesfor the various beauty functionals.

Find good approximations using discrete curvature operators for polyhedra.

Integrate that into an interactive CAD environment.

Minimum Variation CurvesMinimum Variation Curves

They minimize the arc-length integralof the square of the change of curvature.

Very nice overall properties,but depends on global optimization.

A change in one place may affect the shape everywhere !

Local support is preferable... Circle Splines Clothoids

Current Research (Raph Levien)Current Research (Raph Levien)

Clothoids (Euler spirals, Cornu spirals)

for interactive font design (and other CAD)

k s

Clothoid as a General CAD PrimitiveClothoid as a General CAD Primitive

Advantageous features:

Smooth variation of curvature.

Approximates behavior of MVC,but with localized support and control.

Yields circles when allowed by constraints.

Easy calculation of offset curve: clothoid-like.

Raph Levien’s Clothoid Tool KitRaph Levien’s Clothoid Tool Kit

Composite curve of Clothoid segmentsto interpolate given data points

Interactive design toolto draw curves withingiven constraints:position, tangent,and curvature.

Conversion of Clothoidsto Bezier segments withspecified precision. error = 1e-5

ConclusionConclusion

ART

CAD

MATH

Acrobat Reader / Photoshop / FrameMaker

Light Field of Genus-6 Tiffany LampLight Field of Genus-6 Tiffany Lamp

E X T R A SE X T R A S

A Graph-Embedding ProblemA Graph-Embedding Problem

Draw K12 (= complete graph with 12 vertices) crossing-free onto a smooth surface of genus 6 with as much symmetry as possible.

Prof. Bokowski’s ModelsProf. Bokowski’s Models

Physical goose-neck model

Partial virtual model

My Own Rapid-Prototype ModelMy Own Rapid-Prototype Model

Virtual Genus-6 Map (shiny metal)Virtual Genus-6 Map (shiny metal)

Light Field of Genus-6 Tiffany LampLight Field of Genus-6 Tiffany Lamp

SLIDE-UI for “Viae Globi” ShapesSLIDE-UI for “Viae Globi” Shapes

Viae Globi: “Maloja” -- FDM partViae Globi: “Maloja” -- FDM part

A rather winding Swiss mountain pass road in the upper Engadin.

My Agenda within BIDMy Agenda within BID

Teaching Applicable Design Skills

Developing Better CAD Tools & Methods.

Tight coupling: Research Instruction

DoingDoing Design Design In my courses on graphics and on geometrical design, I found that

students often acquire a decent overview of various CAD tools & design techniques – but only at a textbook level.

When asked to design a plastic spoon, they are completely stumped or come up with totally impractical design plans.

My goal is to remedy this situation.

(my)(my) “BID Manifesto” “BID Manifesto”

The proof of any new methods lies in their application to relevant design problems.

Apply new theories & methods of design.

Have the students actually design things – which then (hopefully) prove to be demonstratively better than earlier design.