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Tschirnhausen cubic

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Tschirnhausen cubic

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

Contents

r = a sec 3 ⁡ ( θ / 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 t = tan ⁡ ( θ / 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 ) = a ( 1 − 3 t 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 ) = a t ( 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 .

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

x = 3 a ( 3 − t 2 ) y = a t ( 3 − t 2 )

or

x 3 = 9 a ( x 2 − 3 y 2 ) .

This gives an alternate polar form of

r = 9 a ( sec ⁡ θ − 3 sec ⁡ θ tan 2 ⁡ θ ) .

There is also another equation in Cartesian form that is

3 a y 2 = x ( x − a ) 2 .

References

Tschirnhausen cubic Wikipedia
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