T. J. Peters, UConn Computer Science & Engineering Research & Education Topology & Animation :...

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Transcript of T. J. Peters, UConn Computer Science & Engineering Research & Education Topology & Animation :...

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  • T. J. Peters, UConn Computer Science & Engineering Research & Education Topology & Animation : Science & Technology
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  • Topology (from the Greek , place, and , study)Greek is a major area of mathematics concerned withmathematics spatial properties that are preserved under continuouscontinuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. geometryset theory http://en.wikipedia.org/wiki/Topology Not to be confused with topography.topography
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  • Topology --- Mobius Strip
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  • Topography (from Greek topo-, "place", and graphia, "writing") is the study of Earth's surface shape and features or those of planets, moons, and asteroids. It is also the description of such surface shapes and features (especially their depiction in maps).Earthsurfaceplanetsmoonsasteroids http://en.wikipedia.org/wiki/Topography
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  • Topography Contour lines hiking maps versus Google maps
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  • Stowe Google Maps Different static viewing options
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  • Topology --- Change
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  • Topography Contour lines Static
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  • KnotPlot: www.knotplot.comwww.knotplot.com Unknot or Trefoil? Demo A: Unknown1 & Unknown2
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  • Projection of Knot For a closed curve, c, if there exists some projection such that there are no self-intersections, then c is the unknot. Proof: ?
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  • T. J. Peters, Kerner Graphics & UConn Knots & Molecules in Animation, Simulation & Visualization TEA & ToAST
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  • T. J. Peters Kerner Graphics Topologically Encoded Animation (TEA)
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  • Trefoil Knot 3D Rotation Encode: Rot_0, Rot_1, , Rot_n
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  • More Aggressive Moves Not just rigid body motion Deform shape Preserve crucial characteristics
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  • 1.682 Megs
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  • Homeomorphism is not enough F : X Y, such that F is 1.continuous, 2.1 1 3.onto 4.and has a continuous inverse.
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  • Two Frames with Different Topology
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  • Instantaneous Self-intersection
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  • Contemporary Computational Influences Edelsbrunner: geometry & topology Sethian: Marching methods, topology changes Blackmore: differential sweeps Carlsson, Zomordian : Algebraic
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  • Mappings and Equivalences Knots and self-intersections Piecewise Linear (PL) Approximation My Scientific Emphasis
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  • Isotopy & Animation F : X x [0,1] Y, such that for each t in [0,1] F : X x t is a homeomorphism. We take Y to be 3D space.
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  • Little reuse or modification Plus, we love to blow things up. Kerner Graphics: Digital Visual Effects (DVFX) KERNER OPTICAL
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  • DVFX vs `Blowing things up Modify & re-use vs destroy. But explosions are hard, for now. Provide path for integration.
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  • EagleEye
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  • Moore Dissertation 2006 Efficient algorithm for ambient isotopic PL approximation for Bezier curves of degree 3. Now scale & accelerate.
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  • PL Approximation for Graphics Animation & Visualization (also for Engineeing Design)
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  • Unknot
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  • Bad Approximation! Self-intersect?
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  • Good Approximation! Respects Embedding: Curvature (local) & Separation (global) Error bounds!! => Nbhd_2 about curve. But recognizing unknot in NP (Hass, L, P, 1998)!!
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  • Compression: TEA File (