Tschirnhausen Cubic

In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation
:$r = a\sec^3 \left(\frac\right)$
where is the secant function.

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
:$x=a\cos \theta \sec^3 \frac = a \left(\cos^3 \frac - 3 \cos \frac \sin^2 \frac \right) \sec^3 \frac= a\left(1 - 3 \tan^2 \frac\right)$
::$= a(1 - 3t^2)$
:$y=a\sin \theta \sec^3 \frac = a \left(3 \cos^2 \frac\sin \frac - \sin^3 \frac \right) \sec^3 \frac= a \left(3 \tan \frac - \tan^3 \frac \right)$
::$= at(3-t^2)$
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 8''a'' 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.

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

Plane curves

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