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.