Tschirnhaus'S Cubic

Cartesian equation: 3a y2 = 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

Click THIS LINK to experiment interactively with this curve and its associated curves.

This curve was investigated by Tschirnhaus, de L'Hôpital and Catalan. As well as Tschirnhaus's 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/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Tschirnhaus.html