Tschirnhausen cubic

File:CubiqueTschirnhausen.svg

Tschirnhausen cubic, case of a = 1

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

$r=a{\sec }^3\left(\frac{\theta }{3}\right)$

where sec is the secant function.

History

The curve was studied by Tschirnhaus, de L'Hôpital, and Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by Raymond Clare 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

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

$=a(1-3t^2)$

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

$=at(3-t^2)$

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

$27a{y}^2=(a-x)(8a+x{)}^2$.

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

$x^3=9a(x^2-3y^2)$.

This gives the alternative polar form

$r=9a(\sec \theta -3\sec \theta {\tan }^{}\theta )$.

Generalization

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

References

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

External links

• "Tschirnhaus' Cubic" at MacTutor History of Mathematics archive
• Tschirnhausen cubic at mathcurve.com