Tschirnhausen cubic

In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation where sec is the secant function.

The curve was studied by von Tschirnhaus, de L'Hôpital, and Catalan.

It was given the name Tschirnhausen cubic in a 1900 paper by Raymond Clare Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.

Then applying triple-angle formulas gives giving a parametric form for the curve.

The parameter t can be eliminated easily giving the Cartesian equation If the curve is translated horizontally by 8a and the signs of the variables are changed, the equations of the resulting right-opening curve are and in Cartesian coordinates This gives the alternative polar form The Tschirnhausen cubic is a Sinusoidal spiral with n = −1/3.