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Tschirnhausen cubic

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Plane curve of the form r = a*sec(θ/3)
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 ⁡ ( θ 3 ) {\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}

where sec 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 Raymond Clare Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.

Other equations

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

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

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

27 a y 2 = ( a − x ) ( 8 a + x ) 2 {\displaystyle 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 = 3 a ( 3 − t 2 ) {\displaystyle x=3a(3-t^{2})}
y = a t ( 3 − t 2 ) {\displaystyle y=at(3-t^{2})}

and in Cartesian coordinates

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

This gives the alternative polar form

r = 9 a ( sec ⁡ θ − 3 sec ⁡ θ tan 2 ⁡ θ ) {\displaystyle 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

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