Algebraic geometry on imaginary triangles

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International Electronic Journal of Geometry
Volume:11 Issue:2
Algebraic geometry on imaginary triangles

Algebraic geometry on imaginary triangles

Pages : 71-82

View : 59 | Download : 46

Publication Date : 2018-11-30

Article Type : Research Paper

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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