The stereographic projection of the Stern–Brocot tree

I sketch how the stereographic projection of the Stern–Brocot tree forms an ordered binary tree of Pythagorean triples, which can be used to compute best approximations of turn angles of points on the circle and finally trigonometric functions. Before that, I briefly recapitulate the classical enumeration of Pythagorean triples and the ternary Barning–Hall tree.