Big Picture Defining Barycentric Coordinates From Barycentric Coordinates super simple, you already know them to Whitney Forms: Entending them to Edges, Faces, Tets called Whitney basis functions Turn Your Mesh into a Computational Structure Deriving a whole Discrete Calculus as the one you know, but on mesh Mathieu Desbrun Scared? Don’t be: it’s only numbers on mesh elmts!
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course ACM SIGGRAPH 2005 Course 2
Barycentric Coordinates What’s the Point? You already know barycentric coordinates: Nice, important properties given (d+1) point in Rd, Möbius invented it in 1827 – “mass points” way to locate point within their convex hull ¾ under Gauß’s supervision–got to be good d coordinate-free geometry v0 x ¾ global vs. relative positioning x = bi[x]vi ∑ ¾ i=0 storing numbers on vertices v1 v 1 ¾ intrinsic; indpt on dim. of ambient space d d v inside iff b > 0 ∈ R ¾ affine invariant 2 i x v1 v v with b [x]=1 2 1 v i x ∑ 1 x v v i=0 x 2 0 v v2 v0 v 0 0 Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 3 ACM SIGGRAPH 2005 Course 4
Computing Barycentric Coords Barycentric Basis Functions Question: In 1D: d x = b [x]v Given a d-simplex, find the bary. coords bi [x] ∑ i i of an inside point x for each vertex v such as: i=0 i d vi linear interpolation u(x) = ∑bi[x]ui Unique solution (in any dim.): In 2D: i=0 v1 b [x] = j x v v Try it on a segment 2 0 v0 x Linear Finite Elements… v1 Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 5 ACM SIGGRAPH 2005 Course 6 Extending it to Other Shapes? Generalyzed B. Coords [Warren et al 04] What about polygons/polytopes? For convex polytope, simple expression ? ? ? ? bi[x] = wi[x] / ∑wj[x] , with: v j volume of normal cone Barycentric coordinates for polytopes ouch, not unique anymore… prod. of distances to adj. faces additional requirements? Rational basis fcts of degree (F—n) − smoothness of basis functions (because zero on (F—n) lines) − tensor product (square→bilinear) − face restriction (should fit (n-1)D bc) that extends to smooth domains! − simplicity of evaluation… Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 7 ACM SIGGRAPH 2005 Course 8
Other Types of GBC So What? Other approaches: What did we accomplish? mean value coordinates in 2D and 3D discrete scalar fields ¾ Floater, 2003/2005, positive for any star-shaped polygons ¾ discrete sampling = values on nodes of a mesh Sibson’s – a bit complicated to compute ¾ spatial interpolation to allow arbitrary evals blend of various coordinates Where to Go from Here? ¾ Hormann et al., 2003 what about vector fields? for any 2D polygon [Malsch et al.] what about computations? See new paper by Ju et al. this year ¾ gradient, curl, div, laplacian, you-name-it None reproduces tensor prod. or generalizes to nD keeping them mesh-intrinsic all the way?
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 9 ACM SIGGRAPH 2005 Course 10
Intrinsic Calculus on Meshes Discrete Differential Quantities We can bootstrap a whole discrete calculus! Hinted in the talks before… using only values on simplices they “live” at special places, as distributions and if needed, interpolation in space ¾ Gaussian curvature at vertices ONLY preview: ¾ mean curvature at edges ONLY point-based ∇ edge-based ∇× face-based ∇ • cell-based they can be handled through integration scalar field d vector field d vector field d scalar field ¾ integration calls for k-forms (antisymmetric tensors) deep roots in mathematics − objects that beg to be integrated (ex:∫ f ( x ) dx ) ¾ k-forms are evaluated on kD set ¾ algebraic topology, differential geometry that’s a 1-form − 0-form is evaluated at a point, but very simple to implement and use 1-form at a curve, etc…
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 11 ACM SIGGRAPH 2005 Course 12 Forms You Know For Sure Exterior Calculus of Forms Scalar functions:0-forms Foundation of calculus on smooth manifolds Digital Images: 2-forms Historically, purpose was to extend div/curl/grad incident flux on sensors (W/m2) ¾ Poincaré, Cartan, Lie, … Basis of differential and integral computations Magnetic Field B: 2-form nr r only measurement possible:∫∫ B.nr dA ¾ highlights topological and geometrical structures ¾ modern diff. geometry, Hodge decomposition, … any physical flux is a 2-form too r r r A hierarchy of basic operators are defined: Elctrical Force E: 1-form ∫ E.t dl t ¾ d, *, ∧, b, #, i , L any physical circulation is a 1-form too X X notion of pseudo forms—see notes See [Abraham, Marsden, Ratiu] , ch. 6-7
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 13 ACM SIGGRAPH 2005 Course 14
Discrete Exterior Calculus Discrete Differential Forms Simplicial complex only spatial structure Discrete k-form = values on each kD set ¾ can be 1-, 2-, or 3-manifold, flat or not primal discrete k-form: value on each k-simplex dual discrete k-form: value on each dual (n-k)-cell
0-simplex 1-simplex 2-simplex 3-simplex the rest is defined through linearity dual of k-simplex, Idea: Sampling Forms on Each Simplex! ω = ∑ ω so same cardinality ∫ ∫ 2 1 ¾ “extends” the idea of point-sampling σ j -1.5 U j σ 0.5 j j 0 ¾ use also the “dual” of each simplex in math terms: chains pair with cochains Primal k-simplex (natural pairing = integration) Dual in CS terms: k-form = vector of values Primal Dual k-cell
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 15 ACM SIGGRAPH 2005 Course 16
Notion of Exterior Derivative Exterior Derivative Stokes/Green/{…} theorem: dω = ω Let’s try ∫∫ value? b σσ∂ ∫ dF = F(b) − F(a) a Careful w/ orientation! dF b turns a integral a into a boundary integral ∂ ( )= so d and ∂ are dual ( σ , d ω = ∂ σ , ω ) No “metric” needed! (no size measurement) Implementation? d d d →→→ Try d, then d on an arbitrary form… ¾ As simple as an incidence matrix! ¾ zero; why? ¾ ex: d(1-form)=incidence matrix of edges & faces ¾ because ∂ o ∂ = 0 Bean counting: [|F|x|E|] (|E|) = (|F|) ¾ good: div(curl)=curl(grad)=0 automatically
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 17 ACM SIGGRAPH 2005 Course 18 Hodge Star Discrete deRham Complex Take forms to dual complex (and vice-versa) Discrete calculus through linear algebra: point-based ∇ edge-based ∇× face-based ∇ • cell-based switch values btw primal/dual scalar field vector field vector field scalar field
“diagonal” hodge star cell-based ∇ • face-based ∇× edge-based ∇ point-based scalar field vector field vector field scalar field common “average” simple exercise in matrix assembly ¾ again, a simple (diagonal) matrix value all made out of two trivial operations: now the metric enters… ¾ summing values on simplices (d/∂) Hodge star defines accuracy ¾ scaling values based on local measurements ( ) ¾ order of approximation of the metric Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 19 ACM SIGGRAPH 2005 Course 20
A Step Back Interpolating Discrete Forms Whitney basis fcts to interpolate forms 0-forms (functions) ¾ “hat” functions
1-forms (edge elements!) think ∇ ¾ Whitney forms: for computations ¾ basis of 1-forms since 0 on edges ≠ (i,j) 2-forms: face elmts also a fct of φ and ∇ φ 3-forms: constant per tet Higher order bases for smoother interp. Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 21 ACM SIGGRAPH 2005 Course 22
What We’ll be Able To Do Why DEC is New Not quite like: Finite Elements ¾ use nodes and cells only ¾ tries to enforce local relationships globally well Finite Differences ¾ sorta local polynomial fitting, loses invariants Finite Volumes ¾ use cells and nodes only ¾ good at local relations, often bad at global ones
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 23 ACM SIGGRAPH 2005 Course 24 Why DEC is Limited Why DEC is Good It Does Not Substitute For: If You Ask Me: Numerical Analysis basic discrete operators, consistently derived ¾ accuracy and convergence rate ¾ easy to compute given a discrete mesh still need careful study ¾ separating topology from geometry Good Meshing Tools − helps narrowing down where accuracy is lost ¾ bad mesh? bad results, guaranteed… ¾ conservation laws can be preserved exactly ¾ stay tuned; we’ll address the issue in the last talk preserving structures at the discrete level! Good hacking skills ¾ applicable to a variety of problems ¾ good foundations for further studies ¾ see Building your own DEC at home in notes can be used as basis for ‘simplex sampling’ Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 25 ACM SIGGRAPH 2005 Course 26
Other Related Research Areas Take-Home Message Variational Integrators Geometric Approach to Computations principle of least-action is crucial discrete setup acknowledged from the get-go ¾ motion is a geodesic if we use action as “metric” choice of proper habitat for quantities ¾ preserve invariants and symmetries whole calculus built using only: “not accurate?”: urban legend, simply untrue ¾ boundary of mesh elements Discrete Differential Euclidean Geometry ¾ scaling by local measurements it all ties up through the use of connections preserving structural indentities but it will be for another day… ¾ they are not just abstract concepts: Geometric Algebra they represent defining symmetries
Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction ACM SIGGRAPH 2005 Course 27 ACM SIGGRAPH 2005 Course 28