Tschirnhausen cubic

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

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

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

It is named after the mathematician Ehrenfried Walther von Tschirnhaus.

Other equations

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

$x=a\cos \theta \sec ^{3}{\frac {\theta }{3}}=a(\cos ^{3}{\frac {\theta }{3}}-3\cos {\frac {\theta }{3}}\sin ^{2}{\frac {\theta }{3}})\sec ^{3}{\frac {\theta }{3}}$

$=a(1-3\tan ^{2}{\frac {\theta }{3}})=a(1-3t^{2})$

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

$=a(3\tan {\frac {\theta }{3}}-\tan ^{3}{\frac {\theta }{3}})=at(3-t^{2})$

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

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

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.