Moving Frames and Differential Invariants on Fully Affine Planar Curves


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 to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Brinkman, D., Olver, P.J.: Invariant histograms. Am. Math. Mon. 119, 4–24 (2012)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  3. 3.

    Chou, K.S., Qu, C.Z.: Integrable equations arising from motions of plane curves. Phys. D 162, 9–33 (2002)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey (1976)

    Google Scholar 

  5. 5.

    Fels, M., Olver, P.J.: Moving coframes. I. A practical algorithm. Acta Appl. Math. 51, 161–213 (1998)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Fels, M., Olver, P.J.: Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55, 127–208 (1999)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Hu, N.: Centro-affine space curves with constant curvatures and homogeneous surfaces. J. Geom. 102(1), 103–114 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Hoff, D., Olver, P.J.: Extensions of invariant signatures for object recognition. J. Math. Imaging Vis. 45, 176–185 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kenney, J.P.: Evolution of differential invariant signatures and applications to shape recognition. PhD thesis, University of Minnesota (2009)

  10. 10.

    Kogan, I.A.: Inductive construction of moving frames. Contemp. Math. 285, 157–170 (2001)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kogan, I.A.: Two algorithms for a moving frame construction. Can. J. Math. 55, 266–291 (2003)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Liu, H.L.: Curves in affine and semi-euclidean spaces. Results Math. 65(1), 235–249 (2014)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Nomizu, K., Sasaki, T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  17. 17.

    Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  18. 18.

    Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1, 3–67 (2001)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Olver, P.J.: Moving frames and differential invariants in centro-affine geometry. Lobachevskii J. Math. 31, 77–89 (2010)

    MathSciNet  Article  Google 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)

    Google Scholar 

  24. 24.

    Yang, Y., Yu, Y.: Affine Maurer–Cartan invariants and their applications in self-affine fractals. Fractals 26(4), 1850057 (2018)

    MathSciNet  Article  Google 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 

Download references


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



Corresponding author

Correspondence to Yanhua Yu.

Additional information

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).

Communicated by Young Jin Suh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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).

Download citation


  • Arc length parameter
  • Affine curvature
  • Maurer–Cartan invariant
  • Moving frame

Mathematics Subject Classification

  • 53A15