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Plane curve of the form r = a*sec(θ しーた /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 しーた
3
)
{\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}
where sec is the secant function .
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.
Put
t
=
tan
(
θ しーた
/
3
)
{\displaystyle t=\tan(\theta /3)}
. Then applying triple-angle formulas gives
x
=
a
cos
θ しーた
sec
3
θ しーた
3
=
a
(
cos
3
θ しーた
3
−
3
cos
θ しーた
3
sin
2
θ しーた
3
)
sec
3
θ しーた
3
=
a
(
1
−
3
tan
2
θ しーた
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 しーた
sec
3
θ しーた
3
=
a
(
3
cos
2
θ しーた
3
sin
θ しーた
3
−
sin
3
θ しーた
3
)
sec
3
θ しーた
3
=
a
(
3
tan
θ しーた
3
−
tan
3
θ しーた
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 しーた
−
3
sec
θ しーた
tan
2
θ しーた
)
{\displaystyle r=9a\left(\sec \theta -3\sec \theta \tan ^{2}\theta \right)}
.
The Tschirnhausen cubic is a Sinusoidal spiral with n = −1/3.
J. D. Lawrence, A Catalog of Special Plane Curves . New York: Dover, 1972, pp. 87-90.