Parametrisation of singular curves

The simplest example of a curve with a singularity is typically the semi-cubical parabola. This can be parametrised by intersecting the line through the origin of slope $a$ with the curve $y^2=x^3$ to get $(x,y) = (a^2,a^3)$. More generally, if higher degree curves have enough singularities to make them genus 0 then one can always parameterise them --- however one may need to use higher degree parametrising curves.

Animated parametrisations of singular cubic, quintic and septic curves show the remarkable fact that the only points of intersection of the two curves are at the nodes, cusps and precisely one extra moving point---shown in yellow.

Many thanks to Michael "Immir" Smith for generating the first two cubic plots.