Algebraic geometry on imaginary triangles

Year 2018, Volume: 11 Issue: 2, 71 - 82, 30.11.2018

Sergiy Koshkin

https://doi.org/10.36890/iejg.545133

Abstract

We extend the notion of triangle to imaginary triangles with complex valued sides and angles,

and parametrize families of such triangles by plane algebraic curves.We study in detail families of

triangles with two commensurable angles, and apply the theory of plane Cremona transformations

to find “Pythagorean theorems" for these triangles, which are interpreted as the implicit equations

of their parametrizing curves.

Keywords

Pythagorean triples, Pythagorean theorem, commensurable triangle, Chebyshev polynomials, rational curve, quadratic Cremona transformation, base point, exceptional curvePythagorean triples, exceptional curve

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