Tschirnhausen cubic

In 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).
History.
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.
Other equations.
Put formula_2. 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 8"a" 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 
Generalization.
The Tschirnhausen cubic is a Sinusoidal spiral with "n" = −1/3