Rational hyperbolic triangles and a quartic model of elliptic curves
| Autor: | Jordan Schettler, Nicolas Brody |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Mathematics - Number Theory Plane curve 010102 general mathematics Computer Science::Computational Geometry 01 natural sciences Cubic plane curve Connection (mathematics) Incircle and excircles of a triangle 010101 applied mathematics Elliptic curve Genus (mathematics) Quartic function FOS: Mathematics Number Theory (math.NT) 0101 mathematics Hyperbolic triangle Mathematics |
| Zdroj: | Journal of Number Theory. 164:359-374 |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2016.01.004 |
| Popis: | The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle. Comment: 14 pages, 7 figures |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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