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In geometry , the Tschirnhausen cubic , or Tschirnhaus' cubic]] is a plane curve defined by the polar equation
r
=
a
sec
3
(
θ
/
3
)
.
{\displaystyle r=a\sec ^{3}(\theta /3).}
The Tschirnhausen cubic,
y
2
=
x
3
+
3
x
2
.
{\displaystyle y^{2}=x^{3}+3x^{2}.}
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
)
{\displaystyle t=\tan(\theta /3)}
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
{\displaystyle 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
θ
3
)
=
a
(
1
−
3
t
2
)
{\displaystyle =a\left(1-3\tan ^{2}{\frac {\theta }{3}}\right)=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
{\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
(
3
tan
θ
3
−
tan
3
θ
3
)
=
a
t
(
3
−
t
2
)
{\displaystyle =a\left(3\tan {\frac {\theta }{3}}-\tan ^{3}{\frac {\theta }{3}}\right)=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 then the equations become
x
=
3
a
(
3
−
t
2
)
{\displaystyle x=3a(3-t^{2})}
y
=
a
t
(
3
−
t
2
)
{\displaystyle y=at(3-t^{2})}
or
x
3
=
9
a
(
x
2
−
3
y
2
)
{\displaystyle x^{3}=9a\left(x^{2}-3y^{2}\right)}
This gives an alternate polar form of
r
=
9
a
(
sec
θ
−
3
sec
θ
tan
2
θ
)
{\displaystyle r=9a\left(\sec \theta -3\sec \theta \tan ^{2}\theta \right)}
References
J. D. Lawrence, A Catalog of Special Plane Curves . New York: Dover, 1972, pp. 87-90.
External links
