Moving Frames and Differential Invariants on Fully Affine Planar Curves

  • Yun Yang
  • Yanhua YuEmail author


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.


Arc length parameter Affine curvature Maurer–Cartan invariant Moving frame 

Mathematics Subject Classification




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.


  1. 1.
    Brinkman, D., Olver, P.J.: Invariant histograms. Am. Math. Mon. 119, 4–24 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Chou, K.S., Qu, C.Z.: Integrable equations arising from motions of plane curves. Phys. D 162, 9–33 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey (1976)zbMATHGoogle Scholar
  5. 5.
    Fels, M., Olver, P.J.: Moving coframes. I. A practical algorithm. Acta Appl. Math. 51, 161–213 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fels, M., Olver, P.J.: Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55, 127–208 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu, N.: Centro-affine space curves with constant curvatures and homogeneous surfaces. J. Geom. 102(1), 103–114 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hoff, D., Olver, P.J.: Extensions of invariant signatures for object recognition. J. Math. Imaging Vis. 45, 176–185 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kenney, J.P.: Evolution of differential invariant signatures and applications to shape recognition. PhD thesis, University of Minnesota (2009)Google Scholar
  10. 10.
    Kogan, I.A.: Inductive construction of moving frames. Contemp. Math. 285, 157–170 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kogan, I.A.: Two algorithms for a moving frame construction. Can. J. Math. 55, 266–291 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kogan, I.A., Olver, P.J.: Invariant Euler–Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 76, 137–193 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Liu, H.L.: Curves in affine and semi-euclidean spaces. Results Math. 65(1), 235–249 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mansfield, E., Marí Beffa, G., Wang, J.P.: Discrete moving frames and discrete integrable systems. Found. Comput. Math. 13, 545–582 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Marí Beffa, G., Mansfield, E.L.: Discrete moving frames on lattice varieties and lattice-based multispaces. Found. Comput. Math. 18, 181–247 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nomizu, K., Sasaki, T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  17. 17.
    Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  18. 18.
    Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1, 3–67 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    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)Google Scholar
  20. 20.
    Olver, P.J.: Geometric foundations of numerical algorithms and symmetry. Appl. Alg. Engin. Commun. Comput. 11, 417–436 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Olver, P.J.: Moving frames and differential invariants in centro-affine geometry. Lobachevskii J. Math. 31, 77–89 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Olver, P.J.: Modern developments in the theory and applications of moving frames. Lond. Math. Soc. Impact150 Stories 1, 14–50 (2015)Google Scholar
  23. 23.
    Weyl, H.: Classical Group. Princeton University Press, Princeton (1946)zbMATHGoogle Scholar
  24. 24.
    Yang, Y., Yu, Y.: Affine Maurer–Cartan invariants and their applications in self-affine fractals. Fractals 26(4), 1850057 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yu, G., Morel, J.M.: ASIFT: an algorithm for fully affine invariant comparison. Image Process. On Line 1, 11–38 (2011)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of MathematicsNortheastern UniversityShenyangPeople’s Republic of China
  2. 2.Key Laboratory of Data Analytics and Optimization for Smart Industry (Northeastern University)Ministry of EducationShenyangPeople’s Republic of China

Personalised recommendations