Tschirnhausen cubic

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The Tschirnhausen cubic,

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

History[edit]

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 then the equations become

or

.

This gives an alternate polar form of

.

There is also another equation in Cartesian form that is

.

References[edit]

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

External links[edit]