Tschirnhaus's Cubic

Cartesian equation: 3a y^2 = x(x-a)^2

Click below to see one of the Associated curves.

Definitions of the Associated curves

Evolute       Involute 1       Involute 2

 Inverse curve wrt origin      Inverse wrt another circle

Pedal curve wrt origin      Pedal wrt another point

Negative pedal curve wrt origin      Negative pedal wrt another point

   Caustic wrt horizontal rays      Caustic curve wrt another point
This curve was investigated by Tschirnhaus, de L'Hôpital and Catalan. As well as Tschirnhaus' cubic it is sometimes called de L'Hôpital's cubic or the trisectrix of Catalan.

The name Tschirnhaus's cubic is given in R C Archibald's paper written in 1900 where he attempted to classify curves.

Tschirnhaus's cubic is the negative pedal of a parabola with respect to the focus of the parabola.

The caustic of Tschirnhaus's cubic where the radiant point is the pole is Neile's semi-cubic parabola .

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JOC/EFR February 1996