Primitive Pythagorean Triples on a Curvilinear Grid Defined by Euclid's Formula

A primitive Pythagorean triple is a set of values satisfying the Pythagorean theorem , with no common factors, . The triples can be rewritten using Euclid's formula as with and coprime and odd to ensure that the triples are primitive. The primitive triples are represented by their corresponding triangles, scaled and translated to the intersection of the curves for each value , , in the centroid of the triangle. Empty intersections occur where the triple is not primitive.

Let and . Each blue curve is given parametrically by for fixed ; different values of give different blue curves. The blue curves shown correspond to integer choices of . Similarly, each red curve is given parametrically by for fixed ; different values of give different red curves. The red curves shown correspond to integer choices of .