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.
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