In geometry, Tschirnhausen cubic, is a plane curve defined by the polar equation
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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 then the equations become
or
This gives an alternate polar form of