<i>CR</i>-Selfdual Cubic Curves
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 t...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/2/317 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832588034255618048 |
|---|---|
| author | Mircea Crasmareanu Cristina-Liliana Pripoae Gabriel-Teodor Pripoae |
| author_facet | Mircea Crasmareanu Cristina-Liliana Pripoae Gabriel-Teodor Pripoae |
| author_sort | Mircea Crasmareanu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-fa5861337be94f86958a175c52835bf4 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-fa5861337be94f86958a175c52835bf42025-01-24T13:40:09ZengMDPI AGMathematics2227-73902025-01-0113231710.3390/math13020317<i>CR</i>-Selfdual Cubic CurvesMircea Crasmareanu0Cristina-Liliana Pripoae1Gabriel-Teodor Pripoae2Department of Mathematics, Faculty of Mathematics, “Al.I.Cuza” University, 700506 Iasi, RomaniaDepartment of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana 6, 010374 Bucharest, RomaniaDepartment of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014 Bucharest, RomaniaWe 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.https://www.mdpi.com/2227-7390/13/2/317cubic curvecross ratioCR-selfdual cubicelliptic curveJ-invariantvertices of cubic curves |
| spellingShingle | Mircea Crasmareanu Cristina-Liliana Pripoae Gabriel-Teodor Pripoae <i>CR</i>-Selfdual Cubic Curves Mathematics cubic curve cross ratio CR-selfdual cubic elliptic curve J-invariant vertices of cubic curves |
| title | <i>CR</i>-Selfdual Cubic Curves |
| title_full | <i>CR</i>-Selfdual Cubic Curves |
| title_fullStr | <i>CR</i>-Selfdual Cubic Curves |
| title_full_unstemmed | <i>CR</i>-Selfdual Cubic Curves |
| title_short | <i>CR</i>-Selfdual Cubic Curves |
| title_sort | i cr i selfdual cubic curves |
| topic | cubic curve cross ratio CR-selfdual cubic elliptic curve J-invariant vertices of cubic curves |
| url | https://www.mdpi.com/2227-7390/13/2/317 |
| work_keys_str_mv | AT mirceacrasmareanu icriselfdualcubiccurves AT cristinalilianapripoae icriselfdualcubiccurves AT gabrielteodorpripoae icriselfdualcubiccurves |