Tschirnhausen cubic

In mathematics, the Tschirnhausen cubic is a cubic plane curve defined by the polar equation $${\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}$$ or the equivalent algebraic equation $${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}.}$$

It is a nodal cubic, meaning that it crosses itself at one point, its node. The angle at this crossing point, inside the loop formed by the crossing, is 60°. Because the Tschirnhausen cubic has this singularity, it can be given a parametric equation, and any arc of it can be drawn as a cubic Bézier curve. It is a special case of a sinusoidal spiral, of a pursuit curve, and of a Pythagorean hodograph curve.

The original study of this curve, by Ehrenfried von Tschirnhaus, found it as a caustic of light reflected in a parabolic mirror. It was used by Eugène Catalan in an angle trisection, and it appears among the geodesics of the Enneper surface.