Discrete Differential Geometry: An Applied Introduction
Eitan Grinspun with Mathieu Desbrun & Peter Schröder
DDG Course SIGGRAPH 2005
Why do we care? Q geometry of surfaces Springborn Grape (u. of Bonn) Q mothertongue of physical theories
Q computation: simulation/processing
Alliez et al. Grinspun et al. Desbrun Elcott et al. DDG Course SIGGRAPH 2005
2 A Bit of History
Geometry is the key!
Q studied for centuries Hermann Schwarz, 1890 DiMarco, Physics, Montana Q Cartan, Poincaré, Lie, Hodge, de Rham, Gauss, Noether,… Q mostly differential geometry Q differential and integral calculus The study of invariants and
symmetries Bobenko and Suris DDG Course SIGGRAPH 2005
3 Getting Started
How to apply DiffGeo ideas? Q surfaces as collections of samples Q and topology (connectivity)
DDG Course SIGGRAPH 2005
4 Getting Started
How to apply DiffGeo ideas? Q surfaces as collections of samples Q and topology (connectivity) Q apply continuous ideas Q BUT: setting is discrete Q what is the right way? Q discrete vs. discretized
DDG Course SIGGRAPH 2005
5 Discretized
Build smooth manifold structure Q collection of charts Q mutually compatible on their overlaps Q form an atlas Q realize as smooth functions Q differentiate away… DDG Course SIGGRAPH 2005
6 Discrete Geometry
Basic tool Q differential geometry Hermann Schwarz, 1890 DiMarco, Physics, Montana Q metric, curvature, etc.
Discrete realizations Uli Heller, 2002 Boy’s Surface, Oberwolfach Q “meshes” Q computational geom.
Black Rock City, 2003 Q graph theory Frei Otto, Munich 1968
DDG Course SIGGRAPH 2005
7 Discrete Diff.Geometry
Building from the ground up Q discrete geometry is the given Q meshes: triangles, tets Q more general: cell complex Q how to do calculus? Q pick properties of import
DDG Course SIGGRAPH 2005
8 What Matters?
Structure preservation! Accuracy Speed Q symmetry groups Size Q rigid bodies: Euclidean group Q fluids: diffeomorphism group Q conformal geometry: Möbius group Q many more: symplectic, invariants, Stokes’ theorem, de Rham complex, etc. (pick your favorite…) DDG Course SIGGRAPH 2005
9 Themes for Today
What characterizes structure(s)? Q what is shape? Q Euclidean invariance Q what is physics? Q conservation/balance laws Q what can we measure? Q mass, area, curvature, flux, circulation
DDG Course SIGGRAPH 2005
10 Themes for Today
Invariant descriptions Q quantities invariant under a set of transformations Q symmetries give rise to momenta Intrinsic descriptions Q quantities which do not depend on a coordinate frame
DDG Course SIGGRAPH 2005
11 What it All Means
Benefits Q everything is geometric Q often more straightforward Q tons of indices verboten! The story is not finished… Q still many open questions Q in particular: numerical analysis
DDG Course SIGGRAPH 2005
12 The Program for Today
Things we will cover Q warmup: curves Q discrete analogues of cont. theorems Q surfaces: some basic operators Q the discrete setting Q putting them to work Q denoising/smoothing, parameterization DDG Course SIGGRAPH 2005
13 The Program for Today
Things we will cover Q what can we measure Q invariant measures of “things” Q curvature integrals without derivatives Q a first physics model Q deformation of a shape Q simulating discrete shells
DDG Course SIGGRAPH 2005
14 The Program for Today
Things we will cover Q interpolation on simplicial complexes, i.e., meshes Q discrete exterior calculus Q putting it to work: discrete fluids Q structure preservation: vorticity Q ensured by design!
DDG Course SIGGRAPH 2005
15 The Program for Today
Things we will cover Q conformal geometry Q conformal parameterizations Q curvature energies Q how to make all those meshes Q sampling a surface/volume Q variational tet meshing
DDG Course SIGGRAPH 2005
16