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CR -Selfdual Cubic Curves

Author

Listed:
  • Mircea Crasmareanu

    (Department of Mathematics, Faculty of Mathematics, “Al.I.Cuza” University, 700506 Iasi, Romania
    These authors contributed equally to this work.)

  • Cristina-Liliana Pripoae

    (Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana 6, 010374 Bucharest, Romania
    These authors contributed equally to this work.)

  • Gabriel-Teodor Pripoae

    (Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014 Bucharest, Romania
    These authors contributed equally to this work.)

Abstract

We introduce a special class of cubic curves whose defining parameter satisfies a linear or quadratic equation provided by the values of a cross ratio. There are only seven such cubics and several properties of the real cubics in this class (some of them being elliptic curves) are discussed. Using the Möbius transformation, we extend this self-duality and obtain new families of remarkable complex cubics. In addition, we study (from the differential geometric viewpoint) the surface parameterized by all real cubic curves and we derive its curvature functions. As a by-product, we find a new classification of real Möbius transformations and some estimates for the number of vertices of real cubic curves.

Suggested Citation

  • Mircea Crasmareanu & Cristina-Liliana Pripoae & Gabriel-Teodor Pripoae, 2025. "CR -Selfdual Cubic Curves," Mathematics, MDPI, vol. 13(2), pages 1-23, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:317-:d:1570536
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