Tschirnhausen cubic

In geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined by the polar equation

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 ${\displaystyle t=\tan(\theta /3)}$. 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 then the equations become

or

This gives an alternate polar form of

There is also another equation in Cartesian form that is

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