Imaginary triangles, Pythagorean theorems, and algebraic geometry
Metric Geometry
2017-12-21 v1
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 them, which are interpreted as the implicit equations of their parametrizing curves.
Keywords
Cite
@article{arxiv.1712.07578,
title = {Imaginary triangles, Pythagorean theorems, and algebraic geometry},
author = {Sergiy Koshkin},
journal= {arXiv preprint arXiv:1712.07578},
year = {2017}
}
Comments
15 pages, 1 table, 8 figures