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Discrete Differential Geometry.pdf

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    1
    Discrete Differential Geometry: An Applied Introduction
    Differential Geometry
    A Bit of History
    Getting Started
    Getting Started
    Discretized
    Discrete Geometry
    Discrete Diff.Geometry
    Discr. Diff. Geometry
    What Matters?
    Warmup: Smooth Setting
    Warmup: Smooth Setting
    Warmup: Smooth Setting
    Warmup: Smooth Setting
    Gauß Map:
    Turning Number
    Turning Number Thm.
    Warmup: Discr. Setting
    Inscribed Polygon:
    Length
    Length
    Total Signed Curvature
    Discrete Gauß Map
    Discrete Gauß Map
    Turning Number Theorem
    Structure-Preservation
    Convergence
    Total Signed Curvature
    Another Definition
    Curvature normal
    Gradient of Length
    Gradient of Length
    Gradient of Length
    Gradient of Length
    Gradient of Length
    Gradient of Length
    Gradient of Length
    Moral of the Story
    Themes for Today
    Themes for Today
    What it All Means
    The Program for Today
    The Program for Today
    The Program for Today
    2
    3
    4
    5
    6
    Three Points Make a Triangle… Or a Circle
    In This Section
    Geometries
    Conformal Mappings
    Paramaterizations
    Conformal Maps
    Conformal Maps
    Conformal Maps
    Sounds Great!
    Sounds Great!
    An Old Idea
    An Old Idea
    An Old Idea
    An Old Idea
    History
    Basic Setup
    Circle Pattern Problem
    Circle Pattern Problem
    Circle Pattern Problem
    Geometry at an Edge
    Energy
    Algorithm
    Results
    Boundary Control
    Quasi-Conformal Distr.
    Quasi-Conformal Distr.
    Robustness
    Problems
    Piecewise Flat
    Cone Singularities
    Examples
    Examples
    Properties
    Properties
    Properties
    Summary
    Software
    More Fun with Circles
    More Fun with Circles
    More Fun with Circles
    More Fun with Circles
    Disc. Willmore Energy
    Properties
    Properties
    Properties
    Properties I
    Properties I
    Results I
    Results II
    Results III
    Results III
    Results III
    Results III
    Results III
    Results III
    Results III
    Results III
    Results III
    Results III
    Results IV
    Summary
    Circle Summary
    7
    Discrete Differential Geometry: An Applied Introduction Eitan Grinspun with Mathieu Desbrun, Konrad Polthier, Peter Schröder, & Ari Stern DDG Course SIGGRAPH 2006 1
    Differential Geometry Why do we care? geometry of surfaces mothertongue of physical theories Springborn Grape (u. of Bonn) computation: simulation/processing Elcott et al. Alliez et al. Grinspun et al. Desbrun DDG Course SIGGRAPH 2006 2
    A Bit of History Geometry is the key! studied for centuries Hermann Schwarz, 1890 DiMarco, Physics, Montana Cartan, Poincaré, Lie, Hodge, de Rham, Gauss, Noether,… mostly differential geometry differential and integral calculus The study of invariants and symmetries DDG Course SIGGRAPH 2006 Bobenko and Suris 3
    Getting Started How to apply DiffGeo ideas? surfaces as collections of samples and topology (connectivity) apply continuous ideas BUT: setting is discrete what is the right way? discrete vs. discretized DDG Course SIGGRAPH 2006 4
    Getting Started How to apply DiffGeo ideas? surfaces as collections of samples and topology (connectivity) apply continuous ideas BUT: setting is discrete what is the right way? discrete vs. discretized DDG Course SIGGRAPH 2006 5
    Discretized Build smooth manifold structure collection of charts mutually compatible on their overlaps form an atlas realize as smooth functions differentiate away… DDG Course SIGGRAPH 2006 6
    Discrete Geometry Basic tool differential geometry metric, curvature, etc. Discrete realizations “meshes” computational geom. graph theory DDG Course SIGGRAPH 2006 Hermann Schwarz, 1890 DiMarco, Physics, Montana Uli Heller, 2002 Boy’s Surface, Oberwolfach Black Rock City, 2003 Frei Otto, Munich 1968 7
    Discrete Diff.Geometry Building from the ground up discrete geometry given meshes: triangles, tets more general: cell complex how to do calculus? DDG Course SIGGRAPH 2006 8
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