Tschirnhausen cubic
In geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equationHistory
The curve was studied by von Tschirnhaus, de L'Hôpital, and Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by R C Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.Put. 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
