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Adobe Systems, Strategic Visit, 11/1/06 Artistic Geometry Carlo H. Séquin University of California, Berkeley

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A designer for 30 years…

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Geometry is my true love ! u Optical paths through prism u Integrated circuit layouts u Building floor plans u CAD drawings u Visualizations...

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Fascination with Geometrical Patterns: Escher 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 ? u serious distortions, u not all tiles are the same. Difficulties:

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Infinite Tiling on the Poincaré Disk

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Mapped onto a genus-3 surface 168 auto- morphisms Felix Klein (1878) next: Hurwitz group of order 504

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The 2D Hilbert Curve (1891) A plane-filling Peano curve Fall 1983: CS Graduate Course: Creative Geometric Modeling Do This In 3 D !

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Metal Sculpture at SIGGRAPH 2006

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Since 1994: CAD for Artists u How can we use the visualization power offered by computer graphics and by computer-controlled rapid prototyping for the design of geometrical sculptures?

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Geometric Sculpture Brent Collins

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Art - Math Connection ISAMA & Bridges Conferences MOSAIC 2000 (David Salesin, Univ. of WA)

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Many Math Models and Sculptures CAD tools are still inadequate for free-form 3D designs !

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Ongoing Research Interest CAD tools for Ideation, Informal Prototyping: l Mimick the best of: clay, wire, paper, scotch-tape, styrofoam … l Without the adversity of: messy glue, gravity, strength limits … l 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 …

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Recent Project with Prof. McMains ( ME) Develop a haptic, immersive VR work station for telecollaborative design across the internet See in 3D, Touch, feel, Annotate, Modify, Share, Discuss …

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Research with Prof. Wright (ME) Physical Rapid Prototypes: For early user testing and hands-on feedback in application context. Zcorp 3D-printer FDM From Art to Part Contact-Compact

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Current Research (Pushkar Joshi) l Based on optimization of a functional by using all free DOFs. l Fewer controls to manipulate l Explicit consideration of constraints l Automatic attainment of some quality objectives. Better Surface Design Tools:

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(Future) Surface Optimization Module u Built into future CAD environments! u User specifies all boundary conditions as well as any other constraints. u User selects a suitable beauty functional. u CAD system finds best solution in seconds.

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Beauty Functionals 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

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Pushkar Joshis Work u Build a robust evaluation environment for curvature-based surface functionals. u Create a catalog of minimizing shapes for the various beauty functionals. u Find good approximations using discrete curvature operators for polyhedra. u Integrate that into an interactive CAD environment.

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Minimum Variation Curves u They minimize the arc-length integral of the square of the change of curvature. u Very nice overall properties, but depends on global optimization. u A change in one place may affect the shape everywhere ! u Local support is preferable... Circle Splines Clothoids

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Current Research (Raph Levien) u Clothoids (Euler spirals, Cornu spirals) u for interactive font design (and other CAD) k s

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Clothoid as a General CAD Primitive Advantageous features : u Smooth variation of curvature. u Approximates behavior of MVC, but with localized support and control. u Yields circles when allowed by constraints. u Easy calculation of offset curve: clothoid-like.

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Raph Leviens Clothoid Tool Kit u Composite curve of Clothoid segments to interpolate given data points u Interactive design tool to draw curves within given constraints: position, tangent, and curvature. u Conversion of Clothoids to Bezier segments with specified precision. error = 1e-5

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Conclusion ART CAD MATH Acrobat Reader / Photoshop / FrameMaker

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Light Field of Genus-6 Tiffany Lamp

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E X T R A S

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A Graph-Embedding Problem u Draw K 12 (= complete graph with 12 vertices) crossing-free onto a smooth surface of genus 6 with as much symmetry as possible.

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Prof. Bokowskis Models Physical goose-neck model Partial virtual model

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My Own Rapid-Prototype Model

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Virtual Genus-6 Map (shiny metal)

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Light Field of Genus-6 Tiffany Lamp

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SLIDE-UI for Viae Globi Shapes

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Viae Globi: Maloja -- FDM part u A rather winding Swiss mountain pass road in the upper Engadin.

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My Agenda within BID u Teaching Applicable Design Skills u Developing Better CAD Tools & Methods. Tight coupling: Research Instruction

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Doing Design u 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. u 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.

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(my) BID Manifesto The proof of any new methods lies in their application to relevant design problems. u Apply new theories & methods of design. u Have the students actually design things – which then (hopefully) prove to be demonstratively better than earlier design.

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