From point cloud data to the continuum model of geometry

Transcription

1 From point cloud data to the continuum model of geometry J. Harrison University of California, Berkeley July 22, 2007

2 Continuum model of geometry By the continuum model of geometry we refer to smooth manifolds and associated objects such as smooth functions and differential forms, connections on vector bundles, Riemannian metrics, curvatures and other tensors, and operations like wedge product, Hodge star, exterior derivative, Dirac operator, etc.

3 Continuum model of geometry By the continuum model of geometry we refer to smooth manifolds and associated objects such as smooth functions and differential forms, connections on vector bundles, Riemannian metrics, curvatures and other tensors, and operations like wedge product, Hodge star, exterior derivative, Dirac operator, etc. It involves the plasma of points with calculus connecting things together. With this language of forms, many of the concepts of physics are expressed, but not all.

4 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry:

5 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry: 1. It should be determined by a finite amount of data.

6 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry: 1. It should be determined by a finite amount of data. 2. It should have analogues of all the objects mentioned before.

7 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry: 1. It should be determined by a finite amount of data. 2. It should have analogues of all the objects mentioned before. 3. It should have natural finite dimensional versions which still have all of these structures.

8 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry: 1. It should be determined by a finite amount of data. 2. It should have analogues of all the objects mentioned before. 3. It should have natural finite dimensional versions which still have all of these structures. 4. It should have some fine scale parameter so that when you let this parameter go to zero this discrete model should have a limit which is the smooth continuum.

9 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry: 1. It should be determined by a finite amount of data. 2. It should have analogues of all the objects mentioned before. 3. It should have natural finite dimensional versions which still have all of these structures. 4. It should have some fine scale parameter so that when you let this parameter go to zero this discrete model should have a limit which is the smooth continuum. Sullivan 1998, MSRI streaming video: If this were true you would have algebraic analogues without the problems of analysis. There are famous examples where the analysis cannot be handled.

10 Discrete model of geometry Sullivan articulated four properties for a discrete model of geometry: 1. It should be determined by a finite amount of data. 2. It should have analogues of all the objects mentioned before. 3. It should have natural finite dimensional versions which still have all of these structures. 4. It should have some fine scale parameter so that when you let this parameter go to zero this discrete model should have a limit which is the smooth continuum. Sullivan 1998, MSRI streaming video: If this were true you would have algebraic analogues without the problems of analysis. There are famous examples where the analysis cannot be handled. Poincaré began such a program when he took a space and divided it into cells. This is finite and has a fine scale parameter. It has some of the above objects, but not others.

11 Goals of a discrete model One dreams of using this discrete model to

12 Goals of a discrete model One dreams of using this discrete model to Study finite dimensional analogues of fluid motion and thereby circumvent some famous analytic difficulties.

13 Goals of a discrete model One dreams of using this discrete model to Study finite dimensional analogues of fluid motion and thereby circumvent some famous analytic difficulties. Carry further the discrete regularization of quantum field theories begun by Wilson s lattice gauge theory.

14 Goals of a discrete model One dreams of using this discrete model to Study finite dimensional analogues of fluid motion and thereby circumvent some famous analytic difficulties. Carry further the discrete regularization of quantum field theories begun by Wilson s lattice gauge theory. Assist scientific computation by replacing some of the ad hoc steps in the current algorithms with the universal operations of the discrete model.

15 Goals of a discrete model One dreams of using this discrete model to Study finite dimensional analogues of fluid motion and thereby circumvent some famous analytic difficulties. Carry further the discrete regularization of quantum field theories begun by Wilson s lattice gauge theory. Assist scientific computation by replacing some of the ad hoc steps in the current algorithms with the universal operations of the discrete model. Replace the analytical recipes for characteristic classes and the new 3D and 4D invariants of manifolds by discrete combinatorial algorithms.

16 Goals of a discrete model One dreams of using this discrete model to Study finite dimensional analogues of fluid motion and thereby circumvent some famous analytic difficulties. Carry further the discrete regularization of quantum field theories begun by Wilson s lattice gauge theory. Assist scientific computation by replacing some of the ad hoc steps in the current algorithms with the universal operations of the discrete model. Replace the analytical recipes for characteristic classes and the new 3D and 4D invariants of manifolds by discrete combinatorial algorithms. Develop methods of calculus of variations that permit branched and non orientable solutions, e.g., soap film solutions to Plateau s problem, an open problem of mathematics.

17 Geometrization of forms To help organize and complete the picture, and express more concepts of physics, we propose adding to this collection a new class of objects that are geometric in nature and reverse the variance of differential forms so that the above operators have geometric counterparts such as pushforward, boundary, and Hodge star.

18 Geometrization of forms To help organize and complete the picture, and express more concepts of physics, we propose adding to this collection a new class of objects that are geometric in nature and reverse the variance of differential forms so that the above operators have geometric counterparts such as pushforward, boundary, and Hodge star. Furthermore, we set up a discrete version of the continuum model with analogous objects to all of the above and which has structures preserving finite dimensional truncations. There is a limit of the discrete model which is isomorphic to the continuum model. This discrete model is obtained from a reassembly of information from point cloud data into an infinitesimal algebraic construction that encodes geometry and topology.

19 Currents Currents are higher dimensional versions of distributions.

20 Currents Currents are higher dimensional versions of distributions. Let C k,r (M) denoted the space of differential k-forms on M of class C r with norm ω C r.

21 Currents Currents are higher dimensional versions of distributions. Let C k,r (M) denoted the space of differential k-forms on M of class C r with norm ω C r. Define C k, (M) = lim C k,r (M)

22 Currents Currents are higher dimensional versions of distributions. Let C k,r (M) denoted the space of differential k-forms on M of class C r with norm ω C r. Define C k, (M) = lim C k,r (M) The space of de Rham currents is the topological dual to smooth forms C k, (M) = C k, (M)

23 Currents Currents are higher dimensional versions of distributions. Let C k,r (M) denoted the space of differential k-forms on M of class C r with norm ω C r. Define C k, (M) = lim C k,r (M) The space of de Rham currents is the topological dual to smooth forms C k, (M) = C k, (M) Every object in M that we wish to treat as a domain of integration must be a current as it must act linearly on differential forms. The holy grail has been to find a class of currents with good categorical properties.

24 The search for normed subspaces of currents Laplace Poincaré Subspace Variance Normed /d / operator lemma Currents Covariant no yes yes yes yes

25 The search for normed subspaces of currents Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes

26 The search for normed subspaces of currents Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no

27 The search for normed subspaces of currents Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no

28 The search for normed subspaces of currents Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no

29 The search for normed subspaces of currents Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no

30 Chainlets Recall that currents are defined by C k (M) = (lim C r,k (M))

31 Chainlets Recall that currents are defined by Chainlets are defined by C k (M) = (lim C r,k (M)) C k (M) = lim (C r,k (M) )

32 Chainlets Recall that currents are defined by Chainlets are defined by C k (M) = (lim C r,k (M)) C k (M) = lim (C r,k (M) ) Features

33 Chainlets Recall that currents are defined by Chainlets are defined by C k (M) = (lim C r,k (M)) C k (M) = lim (C r,k (M) ) Features Chainlets represent objects such as submanifolds with cusps, fractals, soap films, graphs of L 1 functions, charged particles...

34 Chainlets Recall that currents are defined by Chainlets are defined by C k (M) = (lim C r,k (M)) C k (M) = lim (C r,k (M) ) Features Chainlets represent objects such as submanifolds with cusps, fractals, soap films, graphs of L 1 functions, charged particles... The limit of operator norms Cr on C r,k (M) is a norm on chainlets called the natural norm = lim r Cr.

35 Chainlets Recall that currents are defined by Chainlets are defined by C k (M) = (lim C r,k (M)) C k (M) = lim (C r,k (M) ) Features Chainlets represent objects such as submanifolds with cusps, fractals, soap films, graphs of L 1 functions, charged particles... The limit of operator norms Cr on C r,k (M) is a norm on chainlets called the natural norm = lim r Cr. Chainlet spaces satisfy a universal property making them the smallest normed subspace of currents for which calculus is valid.

36 Chainlets Recall that currents are defined by Chainlets are defined by C k (M) = (lim C r,k (M)) C k (M) = lim (C r,k (M) ) Features Chainlets represent objects such as submanifolds with cusps, fractals, soap films, graphs of L 1 functions, charged particles... The limit of operator norms Cr on C r,k (M) is a norm on chainlets called the natural norm = lim r Cr. Chainlet spaces satisfy a universal property making them the smallest normed subspace of currents for which calculus is valid. Chainlets contain a discrete subspace as per Sullivan.

37 Pointed chains Let Λ k denote the exterior algebra of k-vectors in R n.

38 Pointed chains Let Λ k denote the exterior algebra of k-vectors in R n. A pointed chain is a finite section of the space of tangent k-vectors of M. That is, a pointed chain P is nonzero except at most finitely many points {p 1,..., p s } called the support of P.

39 Pointed chains Let Λ k denote the exterior algebra of k-vectors in R n. A pointed chain is a finite section of the space of tangent k-vectors of M. That is, a pointed chain P is nonzero except at most finitely many points {p 1,..., p s } called the support of P. We use formal sum notation P = s i=1 (p i; α i ) where α i Λ k.

40 Pointed chains Let Λ k denote the exterior algebra of k-vectors in R n. A pointed chain is a finite section of the space of tangent k-vectors of M. That is, a pointed chain P is nonzero except at most finitely many points {p 1,..., p s } called the support of P. We use formal sum notation P = s i=1 (p i; α i ) where α i Λ k. Let P k (M) denote the space of pointed chains with mass P 0 as a norm.

41 Pointed chains Let Λ k denote the exterior algebra of k-vectors in R n. A pointed chain is a finite section of the space of tangent k-vectors of M. That is, a pointed chain P is nonzero except at most finitely many points {p 1,..., p s } called the support of P. We use formal sum notation P = s i=1 (p i; α i ) where α i Λ k. Let P k (M) denote the space of pointed chains with mass P 0 as a norm. Theorem Pointed chains form a dense subspace of chainlets.

42 Simple elements A simple k-element (p; α) is a pointed chain where p M and α is a simple k-vector.

43 Simple elements A simple k-element (p; α) is a pointed chain where p M and α is a simple k-vector. Geometric operators such as pushforward, exterior product, boundary and geometric Hodge star may be applied to simple k-elements and extended by linearity to bounded operators on pointed chains.

44 Simple elements A simple k-element (p; α) is a pointed chain where p M and α is a simple k-vector. Geometric operators such as pushforward, exterior product, boundary and geometric Hodge star may be applied to simple k-elements and extended by linearity to bounded operators on pointed chains. These operators therefore extend to chainlets without regard to whether or not the chainlet has tangent spaces anywhere defined.

45 Figure: Geometric operators on simple k-elements

46 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no

47 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets

48 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets Covariant

49 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets Covariant yes

50 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets Covariant yes yes

51 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets Covariant yes yes yes

52 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets Covariant yes yes yes yes

53 The table revisited Subspace Variance Normed /d / Laplace Poincaré operator lemma Currents Covariant no yes yes yes yes Forms Contravariant no yes yes yes yes Rectifiable chns Covariant yes no no no no Sharp chains Covariant yes no yes no no Flat chains Covariant yes yes no no no Normal and integral chains Covariant yes yes no no no Chainlets Covariant yes yes yes yes yes

54 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X

55 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X Theorem The natural norm is the largest seminorm on P k (M) such that 1. P P 0

56 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X Theorem The natural norm is the largest seminorm on P k (M) such that 1. P P 0 2. T X P P X 0 P 0 for all P P k (M) and smooth vector fields X.

57 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X Theorem The natural norm is the largest seminorm on P k (M) such that 1. P P 0 2. T X P P X 0 P 0 for all P P k (M) and smooth vector fields X. These two conditions are necessary for calculus, but they are also sufficient.

58 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X Theorem The natural norm is the largest seminorm on P k (M) such that 1. P P 0 2. T X P P X 0 P 0 for all P P k (M) and smooth vector fields X. These two conditions are necessary for calculus, but they are also sufficient. Chainlets are then the smallest normed subspace of currents for which calculus is valid.

59 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X Theorem The natural norm is the largest seminorm on P k (M) such that 1. P P 0 2. T X P P X 0 P 0 for all P P k (M) and smooth vector fields X. These two conditions are necessary for calculus, but they are also sufficient. Chainlets are then the smallest normed subspace of currents for which calculus is valid. They give us the largest dual space of forms and the fewest pathologies for domains.

60 Characterization of the natural norm Let X be a vector field on M and X 0 its sup norm. Let T X denote translation through the time-one map of X Theorem The natural norm is the largest seminorm on P k (M) such that 1. P P 0 2. T X P P X 0 P 0 for all P P k (M) and smooth vector fields X. These two conditions are necessary for calculus, but they are also sufficient. Chainlets are then the smallest normed subspace of currents for which calculus is valid. They give us the largest dual space of forms and the fewest pathologies for domains. If you work with any other subspace of currents, you lose forms, or you lose an operator.

61 Properties Regularity in this category is never a problem because all operators are bounded. You never have to leave the space when you apply an operator.

62 Properties Regularity in this category is never a problem because all operators are bounded. You never have to leave the space when you apply an operator. This theorem provides an equivalent definition of the natural norm from which one may study Cauchy sequences of pointed chains to find many examples of chainlets.

63 Properties Regularity in this category is never a problem because all operators are bounded. You never have to leave the space when you apply an operator. This theorem provides an equivalent definition of the natural norm from which one may study Cauchy sequences of pointed chains to find many examples of chainlets. Pointed chains of arbitrary order and dimension give us a discrete calculus and submanifolds gives us a smooth theory. Since pointed chains and submanifolds are both dense in chainlets, we obtain convergence of the discrete theory of pointed chains to the smooth continuum.

64 Properties Regularity in this category is never a problem because all operators are bounded. You never have to leave the space when you apply an operator. This theorem provides an equivalent definition of the natural norm from which one may study Cauchy sequences of pointed chains to find many examples of chainlets. Pointed chains of arbitrary order and dimension give us a discrete calculus and submanifolds gives us a smooth theory. Since pointed chains and submanifolds are both dense in chainlets, we obtain convergence of the discrete theory of pointed chains to the smooth continuum.

65 Further Properties Define discrete cochains on pointed chains by duality (discovered independently by Arnold, et al, using the Koszul complex) and use this result to characterize the C r norm on cochains using divided differences. Complete to obtain smooth forms using the chainlet isomorphism theorem.

66 Further Properties Define discrete cochains on pointed chains by duality (discovered independently by Arnold, et al, using the Koszul complex) and use this result to characterize the C r norm on cochains using divided differences. Complete to obtain smooth forms using the chainlet isomorphism theorem. Get normal bundles, flux, Stokes, Gauss, Green, vector fields, Lie derivative, creation, annihilation, multiplication by a function (density), calculus of variations,... all with discrete and smooth counterparts.

67 Further Properties Define discrete cochains on pointed chains by duality (discovered independently by Arnold, et al, using the Koszul complex) and use this result to characterize the C r norm on cochains using divided differences. Complete to obtain smooth forms using the chainlet isomorphism theorem. Get normal bundles, flux, Stokes, Gauss, Green, vector fields, Lie derivative, creation, annihilation, multiplication by a function (density), calculus of variations,... all with discrete and smooth counterparts. Theorems are simple to state and prove and are quite general, even optimal.

68 Further Properties Define discrete cochains on pointed chains by duality (discovered independently by Arnold, et al, using the Koszul complex) and use this result to characterize the C r norm on cochains using divided differences. Complete to obtain smooth forms using the chainlet isomorphism theorem. Get normal bundles, flux, Stokes, Gauss, Green, vector fields, Lie derivative, creation, annihilation, multiplication by a function (density), calculus of variations,... all with discrete and smooth counterparts. Theorems are simple to state and prove and are quite general, even optimal. Obtain discrete and smooth concepts of topology and geometry

69 Unification of viewpoints Roughly speaking, the main difference between Finite Element Exterior Calculus (FEEC) and the Calculus of Pointed Chains is that the former discretizes differential forms and the latter discretizes both forms and objects. That is, both integrands and domains of integration are given infinitesimal representations for a full and rich discrete calculus. Meshes and simplicial complexes are examples of chainlets and thus much of Discrete Exterior Calculus (DEC) can be treated within the chainlet viewpoint. The remaining part of the lecture is devoted to two examples. The first gives a simple example where DEC methods as seen in Grinspun, Hirani, Desbrun, and Schröder are simplified and given stability. The second is an important new geometric version of the Poincaré Lemma that relies on discretization of chains, rather than cochains (not available within the framework of FEEC).

70 Chainlets from point cloud data Figure: Underlying manifold

71 Chainlets from point cloud data Figure: Point cloud

72 Chainlets from point cloud data Figure: Delauney triangulation

73 Chainlets from point cloud data Figure: Delauney triangulation

74 Chainlets from point cloud data Figure: Pointed chain approximation to M We may use this pointed chain to formulate discrete versions of the integral theorems of calculus, homology classes and geometry.

75 Discrete unit normal vectors Figure: Discrete unit normals

76 Shape operator Figure: Discrete shape operator The shape operator is the derivative of the Gauss map. It measures the local curvature at a point on a smooth surface.

77 Strain Theorem Let F : M M be a diffeomorphism of surfaces and P and P pointed chain approximations of M and M with P = F P. Let S and S be the shape operators on P and P, respectively. Then Tr(F S ) = F Tr(S ).

78 Strain Theorem Let F : M M be a diffeomorphism of surfaces and P and P pointed chain approximations of M and M with P = F P. Let S and S be the shape operators on P and P, respectively. Then Tr(F S ) = F Tr(S ). We may now approximate strain at p by Tr(F S S) = Tr(F S ) Tr(S) = F Tr(S ) Tr(S).

79 Strain Theorem Let F : M M be a diffeomorphism of surfaces and P and P pointed chain approximations of M and M with P = F P. Let S and S be the shape operators on P and P, respectively. Then Tr(F S ) = F Tr(S ). We may now approximate strain at p by Tr(F S S) = Tr(F S ) Tr(S) = F Tr(S ) Tr(S). Summing over the squares (Tr(F S S)) 2 over all p supp(p), we obtain discrete flexural energy.

80 Stability Our discrete shape operator is a matrix at each point in the point cloud whose determinant and trace approximate Gaussian and mean curvature, respectively. These quantities converge to the these quantities in the smooth continuum by definition of the shape operator in the smooth category and since all operators involved are continuous in the chainlet norm.

81 Stability Our discrete shape operator is a matrix at each point in the point cloud whose determinant and trace approximate Gaussian and mean curvature, respectively. These quantities converge to the these quantities in the smooth continuum by definition of the shape operator in the smooth category and since all operators involved are continuous in the chainlet norm. In particular, discrete flexural energy converges to continuous flexural energy M 4(H F H) 2 da where H and H are mean curvature.

82 Cones in star shaped regions Suppose U is a star-shaped region with respect to q and with diameter R. For p U let J p denote the oriented segment connecting p with q. If α Λ k, let K(p; α) = e α J p. Figure: Cone over a pointed chain

83 Continuity of the cone operator Proposition 1. KP Cr R P Cr ; 2. K K = 0; 3. Id = K + K ; 4. K = ; 5. K K = K.

84 Poincaré Lemma for chainlets Theorem If J is a k-chainlet cycle of class C r supported in a contractible open set U, there exists a (k + 1)-chainlet K of class C r supported in U with J = K. Proof. Set B = KJ. Then B is of class C r by the proposition 1). By part 3) B = KJ = ( K + K )J = J.

85 Algebraic intermediate value theorem Theorem Suppose g : R m R n is a smooth mapping where 1 n m. Suppose J is an n-chainlet in R m and K is an n-chainlet in R n with supp(g J), supp(k) U R n where U is a contractible open set. Then g ( J) = K g J = K.

86 Figure: Algebraic intermediate value theorem

87 Figure: g J = K g J = K

88 General Rolle s Theorem Corollary Suppose Ω is a bounded, open and connected with smooth boundary. If g : Ω R n is smooth and maps Ω to a point q R m, then g has a critical point p 0 Ω. Corollary Let Ω be bounded, open and connected in R n with smooth boundary. If X is a vector field constant on Ω, then X has a critical point in the interior of Ω.

89 General Mean Value Theorem Corollary Suppose Ω is a bounded, open and connected in R n with smooth boundary. If g : Ω R n is smooth and extends to an affine mapping g on Ω, then there exists p 0 Ω such that Dg p0 = g. Proof. By assumption, g extends to an affine mapping g of Ω. Apply Corollary 6 to f = g g. Then f is zero on Ω and thus has a critical point in Ω. The result follows.

90 A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. E.T.Bell Men of Mathematics 1937

91 References D. Arnold, R. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica (2006) E. Grinspun, A. Hirani, M. Desbrun, and P. Schröder, Discrete Shells, Eurographics/SIGGRAPH Symposium on Computer Animation (2003) Desbrun, A. N. Hirani, M. Leok and J. E. Marsden, Discrete exterior calculus. Available from arxiv.org/math.dg/ (2005) J. Harrison, Stokes theorem for nonsmooth chains, Bull. Amer. Math. Soc. 29 (1993) J. Harrison, Ravello lecture notes on Geometric Calculus, (2005) J. Harrison, New geometrical methods in analysis, submitted July, 2007 J. Harrison, Geometric Poincaré Lemma and generalizations of the intermediate and mean value theorems, submitted July, 2007 J. Harrison From Point Cloud Data to Geometry, in preparation.