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Moving Frames and Differential Invariants on Fully Affine Planar Curves

  • Yun Yang
  • Yanhua YuEmail author
Article

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.

Keywords

Arc length parameter Affine curvature Maurer–Cartan invariant Moving frame 

Mathematics Subject Classification

53A15 

Notes

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.

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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

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