# Tschirnhausen cubic

The Tschirnhausen cubic, ${\displaystyle y^{2}=x^{3}+3x^{2}.}$

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

${\displaystyle r=a\sec ^{3}(\theta /3).}$

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

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

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

${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}}$.

If the curve is translated horizontally by 8a then the equations become

${\displaystyle x=3a(3-t^{2})}$
${\displaystyle y=at(3-t^{2})}$

or

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

This gives an alternate polar form of

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

There is also another equation in Cartesian form that is

${\displaystyle 3ay^{2}=x(x-a)^{2}}$.

## References

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