Parameterization of the parabola

It is (well?) known that if \(x = at^2+bt+c\) and \(y=pt^2+qt+r\), then \[ (Ax+By)^2+Cx+Dy+E=0 \] where \[\begin{aligned} A&=p\\ B&=-a\\ C&=qv_2-2pv_1\\ D&=-bv_2+2av_1\\ E&=v_1^2-v_2v_3 \end{aligned}\] with \(\langle v1, v2, v3\rangle =\langle a,b,c\rangle \times \langle p,q,r\rangle\); that is, the \(v_i\) values are the elements of the cross product of the vectors of the coefficients. In other…