Abstract
This paper surveys the new, algorithmic theory of moving frames developed by the author and M. Fels. Applications in geometry, computer vision, classical invariant theory, the calculus of variations, and numerical analysis are indicated.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Akivis, M.A., Rosenfeld, B.A.: Élie Cartan (1869-1951); Translations Math. monographs, vol. 123. American Math. Soc., Providence (1993)
Anderson, I.M.: The Variational Bicomplex, Utah State Technical Report (1989), http://math.usu.edu/~fg_mp
Anderson, I.M.: The Vessiot Handbook, Technical Report, Utah Sate University (2000)
Anderson, I.M., Kamran, N.: The variational bicomplex for second order scalar partial differential equations in the plane. Duke Math. J. 87, 265–319 (1997)
Bazin, P.–L., Boutin, M.: Structure from motion: theoretical foundations of a novel approach using custom built invariants. SIAM J. Appl. Math. 64, 1156–1174 (2004)
Berchenko, I.A., Olver, P.J.: Symmetries of polynomials. J. Symb. Comp. 29, 485–514 (2000)
Bleecker, D.: Gauge Theory and Variational Principles. Addison–Wesley Publ. Co., Reading (1981)
Blumenthal, L.M.: Theory and Applications of Distance Geometry. Oxford Univ. Press, Oxford (1953)
Boutin, M.: Numerically invariant signature curves. Int. J. Computer Vision 40, 235–248 (2000)
Boutin, M.: On orbit dimensions under a simultaneous Lie group action on n copies of a manifold. J. Lie Theory 12, 191–203 (2002)
Boutin, M.: Polygon recognition and symmetry detection. Found. Comput. Math. 3, 227–271 (2003)
Bruckstein, A.M., Holt, R.J., Netravali, A.N., Richardson, T.J.: Invariant signatures for planar shape recognition under partial occlusion. CVGIP: Image Understanding 58, 49–65 (1993)
Bruckstein, A.M., Netravali, A.N.: On differential invariants of planar curves and recognizing partially occluded planar shapes. Ann. Math. Artificial Intel. 13, 227–250 (1995)
Bruckstein, A.M., Rivlin, E., Weiss, I.: Scale space semi-local invariants. Image Vision Comp. 15, 335–344 (1997)
Bruckstein, A.M., Shaked, D.: Skew-symmetry detection via invariant signatures. Pattern Recognition 31, 181–192 (1998)
Budd, C.J., Collins, C.B.: Symmetry based numerical methods for partial differential equations. In: Griffiths, D.F., Higham, D.J., Watson, G.A. (eds.) Numerical analysis 1997. Pitman Res. Notes Math., vol. 380, pp. 16–36. Longman, Harlow (1998)
Budd, C.J., Iserles, A.: Geometric integration: numerical solution of differential equations on manifolds. Phil. Trans. Roy. Soc. London A 357, 945–956 (1999)
Calabi, E., Olver, P.J., Tannenbaum, A.: Affine geometry, curve flows, and invariant numerical approximations. Adv. in Math. 124, 154–196 (1996)
Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Computer Vision 26, 107–135 (1998)
Cantwell, B.J.: Introduction to Symmetry Analysis. Cambridge University Press, Cambridge (2003)
Carlsson, S., Mohr, R., Moons, T., Morin, L., Rothwell, C., Van Diest, M., Van Gool, L., Veillon, F., Zisserman, A.: Semi-local projective invariants for the recognition of smooth plane curves. Int. J. Comput. Vision 19, 211–236 (1996)
Cartan, É.: La Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie, vol. 5. Hermann, Paris (1935)
Cartan, É.: La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitées par la Méthode du Repère Mobile, Cahiers Scientifiques, vol. 18. Gauthier–Villars, Paris (1937)
Cartan, É.: Les sous-groupes des groupes continus de transformations. In: Oeuvres Complètes, part. II, vol. 2, pp. 719–856. Gauthier–Villars, Paris (1953)
Cartan, É.: Sur la structure des groupes infinis de transformations. In: Oeuvres Complètes, part. II, vol. 2, pp. 571–714. Gauthier–Villars, Paris (1953)
Cartan, É.: La structure des groupes infinis. In: Oeuvres Complètes, part. II, vol. 2, pp. 1335–1384. Gauthier–Villars, Paris (1953)
Channell, P.J., Scovel, C.: Symplectic integration of Hamiltonian systems. Nonlinearity 3, 231–259 (1990)
Cheh, J., Olver, P.J., Pohjanpelto, J.: Maurer–Cartan equations for Lie symmetry pseudo-groups of differential equations. J. Math. Phys. (to appear)
Chern, S.-S.: Moving frames. In: Élie Cartan et les Mathématiques d’Aujourn’hui. Soc. Math. France, Astérisque, numéro hors série, pp. 67–77 (1985)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn. Springer, New York (1996)
David, D., Kamran, N., Levi, D., Winternitz, P.: Subalgebras of loop algebras and symmetries of the Kadomtsev-Petviashivili equation. Phys. Rev. Lett. 55, 2111–2113 (1985)
Deeley, R.J., Horwood, J.T., McLenaghan, R.G., Smirnov, R.G.: Theory of algebraic invariants of vector spaces of Killing tensors: methods for computing the fundamental invariants. Proc. Inst. Math. NAS Ukraine 50, 1079–1086 (2004)
Dhooghe, P.F.: Multilocal invariants. In: Dillen, F., Komrakov, B., Simon, U., Van de Woestyne, I., Verstraelen, L. (eds.) Geometry and Topology of Submanifolds, vol. VIII, pp. 121–137. World Sci. Publishing, Singapore (1996)
Dorodnitsyn, V.A.: Transformation groups in net spaces. J. Sov. Math. 55, 1490–1517 (1991)
Dorodnitsyn, V.A.: Finite difference models entirely inheriting continuous symmetry of original differential equations. Int. J. Mod. Phys. C 5, 723–734 (1994)
Ehresmann, C.: Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie. In: Géometrie Différentielle, Strasbourg. Colloq. Inter. du Centre Nat. de la Rech. Sci., pp. 97–110 (1953)
Fefferman, C., Graham, C.R.: Conformal invariants. In: Élie Cartan et les Mathématiques d’aujourd’hui, France, Paris. Astérisque, hors série. Soc. Math., pp. 95–116 (1985)
Faugeras, O.: Cartan’s moving frame method and its application to the geometry and evolution of curves in the euclidean, affine and projective planes. In: Mundy, J.L., Zisserman, A., Forsyth, D. (eds.) AICV 1993. LNCS, vol. 825, pp. 11–46. Springer, Heidelberg (1994)
Fels, M., Olver, P.J.: Moving coframes. I. A practical algorithm. Acta Appl. Math. 51, 161–213 (1998)
Fels, M., Olver, P.J.: Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55, 127–208 (1999)
Fuchs, D.B., Gabrielov, A.M., Gel’fand, I.M.: The Gauss–Bonnet theorem and Atiyah–Patodi–Singer functionals for the characteristic classes of foliations. Topology 15, 165–188 (1976)
Green, M.L.: The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces. Duke Math. J. 45, 735–779 (1978)
Greene, B.: The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. W.W. Norton, New York (1999)
Griffiths, P.A.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41, 775–814 (1974)
Griffiths, P.A.: Exterior Differential Systems and the Calculus of Variations. In: Progress in Math., vol. 25. Birkhäuser, Boston (1983)
Guggenheimer, H.W.: Differential Geometry. McGraw–Hill, New York (1963)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, New York (2002)
Hann, C.E., Hickman, M.S.: Projective curvature and integral invariants. Acta Appl. Math. 74, 177–193 (2002)
Hilbert, D.: Theory of Algebraic Invariants. Cambridge Univ. Press, New York (1993)
Hubert, E.: Differential algebra for derivations with nontrivial commutation rules, preprint, INRIA Research Report 4972, Sophis Antipolis, France (2003)
Itskov, V.: Orbit Reduction of Exterior Differential Systems, Ph. D. Thesis, University of Minnesota (2002)
Jaegers, P.J.: Lie group invariant finite difference schemes for the neutron diffusion equation. Ph. D. Thesis, Los Alamos National Lab Report, LA–12791–T (1994)
Jensen, G.R.: Higher order contact of submanifolds of homogeneous spaces. Lecture Notes in Math., vol. 610. Springer, New York (1977)
Kemper, G., Boutin, M.: On reconstructing n-point configurations from the distribution of distances or areas. Adv. App. Math. 32, 709–735 (2004)
Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Arch. Rat. Mech. Anal. 134, 275–301 (1996)
Killing, W.: Erweiterung der Begriffes der Invarianten von Transformationgruppen. Math. Ann. 35, 423–432 (1890)
Kim, P., Olver, P.J.: Geometric integration via multi-space. Regular and Chaotic Dynamics 9, 213–226 (2004)
Kogan, I.A.: Inductive approach to moving frames and applications in classical invariant theory, Ph. D. Thesis, University of Minnesota (2000)
Kogan, I.A., Moreno Maza, M.: Computation of canonical forms for ternary cubics. In: Mora, T. (ed.) Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, The Association for Computing Machinery, New York. The Association for Computing Machinery, pp. 151–160 (2002)
Kogan, I.A.: Personal Communication (2004)
Kogan, I.A., Olver, P.J.: The invariant variational bicomplex. Contemp. Math. 285, 131–144 (2001)
Kogan, I.A., Olver, P.J.: Invariant Euler-Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 76, 137–193 (2003)
Kumpera, A.: Invariants différentiels d’un pseudogroupe de Lie. J. Diff. Geom. 10, 289–416 (1975)
Kuranishi, M.: On the local theory of continuous infinite pseudo groups I. Nagoya Math. J. 15, 225–260 (1959)
Lewis, D., Simo, J.C.: Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups. J. Nonlin. Sci. 4, 253–299 (1994)
Lie, S.: Über unendlichen kontinuierliche Gruppen. In: Christ. Forh. Aar., vol. 8, pp. 1–47 (1883); Also Gesammelte Abhandlungen, vol. 5, pp. 314–360. B.G. Teubner, Leipzig (1924)
Lie, S.: Über Integralinvarianten und ihre Verwertung für die Theorie der Differentialgleichungen. Leipz. Berichte 49, 369–410 (1897); Also Gesammelte Abhandlungen, vol. 6, pp. 664–701. B.G. Teubner, Leipzig (1927)
Mackenzie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry. London Math. Soc. Lecture Notes, vol. 124. Cambridge University Press, Cambridge (1987)
Mansfield, E.L.: Algorithms for symmetric differential systems. Found. Comput. Math. 1, 335–383 (2001)
Mansfield, E.L.: Personal Communication (2003)
Marí Beffa, G.: Relative and absolute differential invariants for conformal curves. J. Lie Theory 13, 213–245 (2003)
Marí–Beffa, G., Olver, P.J.: Differential invariants for parametrized projective surfaces. Commun. Anal. Geom. 7, 807–839 (1999)
McLachlan, R.I., Quispel, G.R.W.: Six lectures on the geometric integration of ODEs. In: DeVore, R., Iserles, A., Suli, E. (eds.) Foundations of Computational Mathematics. London Math. Soc. Lecture Note Series, vol. 284, pp. 155–210. Cambridge University Press, Cambridge (2001)
McLachlan, R.I., Quispel, G.R.W.: What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration. Nonlinearity 14, 1689–1705 (2001)
McLenaghan, R.G., Smirnov, R.G., The, D.: An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics. J. Math. Phys. 45, 1079–1120 (2004)
Medolaghi, P.: Classificazione delle equazioni alle derivate parziali del secondo ordine, che ammettono un gruppo infinito di trasformazioni puntuali. Ann. Mat. Pura Appl. 1(3), 229–263 (1898)
Menger, K.: Untersuchungen über allgemeine Metrik. Math. Ann. 100, 75–163 (1928)
Milne–Thompson, L.M.: The Calculus of Finite Differences. Macmilland and Co., Ltd., London (1951)
Moons, T., Pauwels, E., Van Gool, L., Oosterlinck, A.: Foundations of semi-differential invariants. Int. J. Comput. Vision 14, 25–48 (1995)
Morozov, O.: Moving coframes and symmetries of differential equations. J. Phys. A 35, 2965–2977 (2002)
Olver, P.J.: Classical invariant theory and the equivalence problem for particle Lagrangians. I. Binary Forms. Adv. in Math. 80, 39–77 (1990)
Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Graduate Texts in Mathematics, vol. 107. Springer, New York (1993)
Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)
Olver, P.J.: Classical Invariant Theory. London Math. Soc. Student Texts, vol. 44. Cambridge University Press, Cambridge (1999)
Olver, P.J.: Moving frames and singularities of prolonged group actions. Selecta Math. 6, 41–77 (2000)
Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1, 3–67 (2001)
Olver, P.J.: On multivariate interpolation. University of Minnesota (2003)
Olver, P.J.: Geometric foundations of numerical algorithms and symmetry. Appl. Alg. Engin. Commun. Comput. 11, 417–436 (2001)
Olver, P.J., Pohjanpelto, J.: Moving frames for pseudo–groups. I. The Maurer–Cartan forms, preprint, University of Minnesota (2004)
Olver, P.J., Pohjanpelto, J.: Moving frames for pseudo–groups. II. Differential invariants for submanifolds, preprint, University of Minnesota (2004)
Olver, P.J., Pohjanpelto, J.: Regularity of pseudogroup orbits. In: Gaeta, G. (ed.) Symmetry and Perturbation Theory (to appear)
Pauwels, E., Moons, T., Van Gool, L.J., Kempenaers, P., Oosterlinck, A.: Recognition of planar shapes under affine distortion. Int. J. Comput. Vision 14, 49–65 (1995)
Powell, M.J.D.: Approximation theory and Methods. Cambridge University Press, Cambridge (1981)
Robart, T., Kamran, N.: Sur la théorie locale des pseudogroupes de transformations continus infinis I. Math. Ann. 308, 593–613 (1997)
Rund, H.: The Hamilton-Jacobi Theory in the Calculus of Variations. D. Van Nostrand Co. Ltd., Princeton (1966)
Salmon, R.: Lectures on Geophysical Fluid Dynamics. Oxford Univ. Press, Oxford (1998)
Shakiban, C., Lloyd, P.: Classification of signature curves using latent semantic analysis. University of St. Thomas (2004)
Shokin, Y.I.: The Method of Differential Approximation. Springer, New York (1983)
Singer, I.M., Sternberg, S.: The infinite groups of Lie and Cartan. Part I (the transitive groups). J. Analyse Math. 15, 1–114 (1965)
Sokolov, V.V., Zhiber, A.V.: On the Darboux integrable hyperbolic equations. Phys. Lett. A 208, 303–308 (1995)
Spencer, D.C.: Deformations of structures on manifolds defined by transitive pseudo-groups I, II. Ann. Math. 76, 306–445 (1962)
Tresse, A.: Sur les invariants différentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)
Tsujishita, T.: On variational bicomplexes associated to differential equations. Osaka J. Math. 19, 311–363 (1982)
van Beckum, F.P.H., van Groesen, E.: Discretizations conserving energy and other constants of the motion. In: Proc. ICIAM 1987, Paris, pp. 17–35 (1987)
Van Gool, L., Brill, M.H., Barrett, E.B., Moons, T., Pauwels, E.: Semi-differential invariants for nonplanar curves. In: Mundy, J.L., Zisserman, A. (eds.) Geometric Invariance in Computer Vision, pp. 293–309. MIT Press, Cambridge (1992)
Van Gool, L., Moons, T., Pauwels, E., Oosterlinck, A.: Semi-differential invariants. In: Mundy, J.L., Zisserman, A. (eds.) Geometric Invariance in Computer Vision, pp. 157–192. MIT Press, Cambridge (1992)
Weinstein, A.: Groupoids: unifying internal and external symmetry. A tour through some examples. Contemp. Math. 282, 1–19 (2001)
Weyl, H.: Classical Groups. Princeton Univ. Press, Princeton (1946)
Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE Trans. Medical Imaging 16, 199–209 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Olver, P.J. (2005). A Survey of Moving Frames. In: Li, H., Olver, P.J., Sommer, G. (eds) Computer Algebra and Geometric Algebra with Applications. IWMM GIAE 2004 2004. Lecture Notes in Computer Science, vol 3519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11499251_11
Download citation
DOI: https://doi.org/10.1007/11499251_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26296-1
Online ISBN: 978-3-540-32119-4
eBook Packages: Computer ScienceComputer Science (R0)