Tschirnhausen cubic

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Tschirnhausen cubic, case of a = 1

In geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, 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.

Other equations[edit]

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



  • J. D. Lawrence, A Catalog of Special Plane Curves. New York: Dover, 1972, pp. 87-90.

External links[edit]

  • Weisstein, Eric W. "Tschirnhausen Cubic". MathWorld.
  • "Tschirnhaus' Cubic" at MacTutor History of Mathematics Archive
  • Tschirnhausen cubic at mathcurves.com