Four point parabolas

Introduction It is (or should be) well known that a parabola has the cartesian form \[ (Ax+By)^2+Cx+Dy+E = 0. \] This looks as though there are five values needed, but we can divide through in such a way as to make any of the coefficients we like equal to 1: \[ (Px+Qy)^2+Rx+Sy+1 = 0. \] and so we see that only four values are needed to define a parabola. A parabola thus has one more degree of…