Affine differential invariants of functions on the plane. (English) Zbl 1271.53013

Summary: A differential invariant is a function defined on the jet space of functions that remains the same under a group action. It is an important concept to solve the equivalence problem. This paper presents an effective method to derive a special type of affine differential invariants. Given some functions defined on the plane and an affine group acting on the plane, there are induced actions of the group on the functions and on the derivative functions of the functions. Affine differential invariants of these functions are useful in many applications. However, there has been little systematic study of this problem at present. No clear and simple results are available for application users to use directly. We propose a direct and simple method to construct affine differential invariants in this situation. Some useful explicit formulas of affine differential invariants of 2D functions are presented.


53A55 Differential invariants (local theory), geometric objects
22F05 General theory of group and pseudogroup actions
Full Text: DOI


[1] D. Hilbert, Theory of Algebraic Invariants, Cambridge University Press, Cambridge, Mass, USA, 1993. · Zbl 0801.13001
[2] J. H. Grace and A. Young, The Algebra of Invariants, Cambridge University Press, Cambridge, Mass, USA, 1903. · Zbl 1249.05300
[3] O. E. Glenn, A Treatise on the Theory of Invariants, Ginn and Company, Boston, Mass, USA, 1915. · JFM 45.0240.01
[4] P. J. Olver, Classical Invariant Theory, vol. 44, Cambridge University Press, Cambridge, Mass, USA, 1999. · Zbl 1053.33003
[5] S. Lie, “Über Differentialinvarianten,” Mathematische Annalen, vol. 24, no. 4, pp. 537-578, 1884. · JFM 16.0091.01
[6] A. Tresse, “Sur les invariants différentiels des groupes continus de transformations,” Acta Mathematica, vol. 18, no. 1, pp. 1-88, 1894. · JFM 25.0641.01
[7] J. L. Mundy and A. Zisserman, Geometric Invariance in Computer Vision, MIT Press, Cambridge, Mass, USA, 1992.
[8] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, Cambridge, Mass, USA, 2nd edition, 2003. · Zbl 0956.68149
[9] Y. B. Wang, B. Zhang, and T. S. Yao, “Projective invariants of co-moments of 2D images,” Pattern Recognition, vol. 43, pp. 3233-3242, 2010. · Zbl 1205.68486
[10] É. Cartan, La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés, Hermann & Cie, Paris, France, 1935. · Zbl 0010.39501
[11] É. Cartan, La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitees par la Méthode du Repère Mobile, Gautier-Villars, Paris, France, 1937. · Zbl 0018.29804
[12] M. Fels and P. J. Olver, “Moving coframes-II. Regularization and theoretical foundations,” Acta Applicandae Mathematicae, vol. 55, no. 2, pp. 127-208, 1999. · Zbl 0937.53013
[13] M. Fels and P. J. Olver, “Moving coframes-I. A practical algorithm,” Acta Applicandae Mathematicae, vol. 51, no. 2, pp. 161-213, 1998. · Zbl 0937.53012
[14] E. Hubert and I. A. Kogan, “Smooth and algebraic invariants of a group action: local and global constructions,” Foundations of Computational Mathematics, vol. 7, no. 4, pp. 455-493, 2007. · Zbl 1145.53006
[15] E. Hubert, “Differential invariants of a Lie group action: syzygies on a generating set,” Journal of Symbolic Computation, vol. 44, no. 4, pp. 382-416, 2009. · Zbl 1176.12004
[16] I. A. Kogan and P. J. Olver, “Invariant Euler-Lagrange equations and the invariant variational bicomplex,” Acta Applicandae Mathematicae, vol. 76, no. 2, pp. 137-193, 2003. · Zbl 1034.53015
[17] P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, Mass, USA, 1995. · Zbl 0837.58001
[18] P. J. Olver, “Differential invariants of surfaces,” Differential Geometry and Its Applications, vol. 27, no. 2, pp. 230-239, 2009. · Zbl 1164.53002
[19] P. J. Olver, “Generating differential invariants,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 450-471, 2007. · Zbl 1124.53006
[20] P. J. Olver and J. Pohjanpelto, “Differential invariant algebras of Lie pseudo-groups,” Advances in Mathematics, vol. 222, no. 5, pp. 1746-1792, 2009. · Zbl 1194.58018
[21] P. J. Olver, “Moving frames and differential invariants in centro-affine geometry,” Lobachevskii Journal of Mathematics, vol. 31, no. 2, pp. 77-89, 2010. · Zbl 1260.53024
[22] A. V. Aminova and N. A.-M. Aminov, “Projective geometry of systems of second-order differential equations,” Rossiĭskaya Akademiya Nauk. Matematicheskiĭ Sbornik, vol. 197, no. 7, article 951, 2006. · Zbl 1143.53310
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