Tschirnhausen Cubic

The Tschirnhausen cubic is a plane curve given by the polar equation

(1)

Letting gives the parametric equations

			(2)
			(3)

or

			(4)
			(5)

(Lawrence 1972, p. 88).

Eliminating from the above equations gives the Cartesian equations

			(6)
			(7)

(Lawrence 1972, p. 88).

The curve is also known as Catalan's trisectrix and l'Hospital's cubic. The name Tschirnhaus's cubic is given in R. C. Archibald's 1900 paper attempting to classify curves (MacTutor Archive).

The curve has a loop, illustrated above, corresponding to in the above parametrization. The area of the loop is given by

			(8)
			(9)
			(10)
			(11)

(Lawrence 1972, p. 89).

In the first parametrization, the arc length, curvature, and tangential angle as a function of are

			(12)
			(13)
			(14)

The curve has a single ordinary double point located at in the parametrization of equations (◇) and (◇).

The Tschirnhausen cubic is the negative pedal curve of a parabola with respect to the focus and the catacaustic of a parabola with respect to a point at infinity perpendicular to the symmetry axis.