Tschirnhausen cubic

Last updated June 04, 2023

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

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 $t=\tan(\theta /3)$ . Then applying triple-angle formulas gives

x=a\cos \theta \sec ^{3}{\frac {\theta }{3}}=a\left(\cos ^{3}{\frac {\theta }{3}}-3\cos {\frac {\theta }{3}}\sin ^{2}{\frac {\theta }{3}}\right)\sec ^{3}{\frac {\theta }{3}}=a\left(1-3\tan ^{2}{\frac {\theta }{3}}\right)

=a(1-3t^{2})

y=a\sin \theta \sec ^{3}{\frac {\theta }{3}}=a\left(3\cos ^{2}{\frac {\theta }{3}}\sin {\frac {\theta }{3}}-\sin ^{3}{\frac {\theta }{3}}\right)\sec ^{3}{\frac {\theta }{3}}=a\left(3\tan {\frac {\theta }{3}}-\tan ^{3}{\frac {\theta }{3}}\right)

=at(3-t^{2})

giving a parametric form for the curve. The parameter t can be eliminated easily giving the Cartesian equation

27ay^{2}=(a-x)(8a+x)^{2}

.

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

x=3a(3-t^{2})

y=at(3-t^{2})

and in Cartesian coordinates

x^{3}=9a\left(x^{2}-3y^{2}\right)

.

This gives the alternative polar form

r=9a\left(\sec \theta -3\sec \theta \tan ^{2}\theta \right)

.

Generalization

The Tschirnhausen cubic is a Sinusoidal spiral with n = −1/3.

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References

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

External links

Weisstein, Eric W. "Tschirnhausen Cubic". MathWorld .
"Tschirnhaus' Cubic" at MacTutor History of Mathematics archive
Tschirnhausen cubic at mathcurve.com

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