CR -Selfdual Cubic Curves

Mircea Crasmareanu, Cristina-Liliana Pripoae and Gabriel-Teodor Pripoae ()
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Mircea Crasmareanu: Department of Mathematics, Faculty of Mathematics, “Al.I.Cuza” University, 700506 Iasi, Romania
Cristina-Liliana Pripoae: Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana 6, 010374 Bucharest, Romania
Gabriel-Teodor Pripoae: Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014 Bucharest, Romania

Mathematics, 2025, vol. 13, issue 2, 1-23

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.

Keywords: cubic curve; cross ratio; CR-selfdual cubic; elliptic curve; J-invariant; vertices of cubic curves; classification of Möbius transformations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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