Abstract
In this paper, by the affine analogue of the fundamental theorem for Euclidean planar curves, we classify the affine curves with constant affine curvatures. Note that we use the fully affine group and not the equi-affine subgroup consisting of area-preserving affine transformations. (Caution: much of the literature omits the “equi-” in their treatment.) According to the equivariant method of moving frames, explicit formulas for the generating affine differential invariants and invariant differential operators are constructed. At the same time, by using the fact that the affine transformation group GA\((2,\mathbb {R})\) can factor as a product of two subgroup \(B\cdot \mathrm{SE}(2,\mathbb {R})\) and the moving frame of the subgroup SE\((2,\mathbb {R})\), we build the moving frame of GA\((2,\mathbb {R})\) and obtain the relations among invariants of group GA\((2,\mathbb {R})\) and its subgroup SE\((2,\mathbb {R})\). Applying the affine curvature to recognize affine equivalent objects is considered in the last part of this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brinkman, D., Olver, P.J.: Invariant histograms. Am. Math. Mon. 119, 4–24 (2012)
Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26, 107–135 (1998)
Chou, K.S., Qu, C.Z.: Integrable equations arising from motions of plane curves. Phys. D 162, 9–33 (2002)
Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey (1976)
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)
Hu, N.: Centro-affine space curves with constant curvatures and homogeneous surfaces. J. Geom. 102(1), 103–114 (2011)
Hoff, D., Olver, P.J.: Extensions of invariant signatures for object recognition. J. Math. Imaging Vis. 45, 176–185 (2013)
Kenney, J.P.: Evolution of differential invariant signatures and applications to shape recognition. PhD thesis, University of Minnesota (2009)
Kogan, I.A.: Inductive construction of moving frames. Contemp. Math. 285, 157–170 (2001)
Kogan, I.A.: Two algorithms for a moving frame construction. Can. J. Math. 55, 266–291 (2003)
Kogan, I.A., Olver, P.J.: Invariant Euler–Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 76, 137–193 (2003)
Liu, H.L.: Curves in affine and semi-euclidean spaces. Results Math. 65(1), 235–249 (2014)
Mansfield, E., Marí Beffa, G., Wang, J.P.: Discrete moving frames and discrete integrable systems. Found. Comput. Math. 13, 545–582 (2013)
Marí Beffa, G., Mansfield, E.L.: Discrete moving frames on lattice varieties and lattice-based multispaces. Found. Comput. Math. 18, 181–247 (2018)
Nomizu, K., Sasaki, T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)
Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1999)
Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1, 3–67 (2001)
Olver, P.J.: Moving frames-in geometry, algebra, computer vision, and numerical analysis. In: DeVore, R., Iserles, A., Suli, E. (eds.) Foundations of Computational Mathematics, volume 284 of London Mathematical Society. Lecture Note Series, pp. 67–297. Cambridge University Press, Cambridge (2001)
Olver, P.J.: Geometric foundations of numerical algorithms and symmetry. Appl. Alg. Engin. Commun. Comput. 11, 417–436 (2001)
Olver, P.J.: Moving frames and differential invariants in centro-affine geometry. Lobachevskii J. Math. 31, 77–89 (2010)
Olver, P.J.: Modern developments in the theory and applications of moving frames. Lond. Math. Soc. Impact150 Stories 1, 14–50 (2015)
Weyl, H.: Classical Group. Princeton University Press, Princeton (1946)
Yang, Y., Yu, Y.: Affine Maurer–Cartan invariants and their applications in self-affine fractals. Fractals 26(4), 1850057 (2018)
Yu, G., Morel, J.M.: ASIFT: an algorithm for fully affine invariant comparison. Image Process. On Line 1, 11–38 (2011)
Acknowledgements
The authors wish to express the utmost sincere thanks to Prof. Peter J. Olver for hosting the first author as a visitor at the University of Minnesota and ongoing discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Young Jin Suh.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by China Scholarship Council (Grant No. 201706085065), the Fundamental Research Funds for the Central Universities (Grant No. N170504014) , 111 Project (Grant No. B16009), the Fund for Innovative Research Groups of the National Natural Science Foundation of China (71621061) and the Major International Joint Research Project of the National Natural Science Foundation of China (71520107004).
Rights and permissions
About this article
Cite this article
Yang, Y., Yu, Y. Moving Frames and Differential Invariants on Fully Affine Planar Curves. Bull. Malays. Math. Sci. Soc. 43, 3229–3258 (2020). https://doi.org/10.1007/s40840-019-00864-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-019-00864-z