Tschirnhausen cubic
In mathematics, the Tschirnhausen cubic is a cubic plane curve defined in Cartesian coordinates ( x , y ) {\displaystyle (x,y)} by the cubic equation 27 a y 2 = ( a − x ) ( 8 a + x ) 2 , {\displaystyle 27ay^{2}=(a-x)(8a+x)^{2},} or in polar coordinates ( r , θ ) {\displaystyle (r,\theta )} by the equivalent trigonometric equation involving the secant function, r = a sec 3 θ 3 . {\displaystyle r=a\sec ^{3}{\frac {\theta }{3}}.} It is a nodal cubic, meaning that it crosses itself at one point, its node.