Parametrization of Triangles by Elliptic Curves
- Jason Murphy
We begin by posing a seemingly innocent question from Euclidean geometry: if two triangles have the same area and perimeter, are they necessarily congruent? The answer to this question is no, but the more interesting part of this answer is that all triangles sharing the same perimeter and area can be parametrized by points on a particular family of elliptic curves (over a suitably defined field). In particular, let and suppose . Then any triangle with area and perimeter determines a point on the nonsingular cubic (and, thus, elliptic) curve: There is a natural group law for the curve , which is not unlike that of elliptic curves in Weierstrass form: take any two distinct points and draw the line between them, which will be secant to . By Bezout's Theorem, this line will intersect the curve in precisely one more point, which we call Reflecting this point over the line in the affine real plane then yields the point . If this point lies in the first quadrant, it will define (up to a similarity transformation) a new triangle with area and perimeter . In the figure below, we illustrate the group law on the triangle curve (shown on the left) and its corresponding curve in short Weierstrass form (shown on the right): By moving the points and around and changing the area and semiperimeter , see what new triangles you can generate!