Bicorn

The bicorn, sometimes also called the "cocked hat curve" (Cundy and Rollett 1989, p. 72), is the name of a collection of quartic curves studied by Sylvester in 1864 and Cayley in 1867 (MacTutor Archive). The bicorn is given by the parametric equations

			(1)
			(2)

(Lawrence 1972, p. 147) and Cartesian equation

(3)

(Lawrence 1972, p. 147; Cundy and Rollett 1989, p. 72; Mactutor, with the final corrected to be squared instead of to the first power).

The bicorn has cusps at .

The area enclosed by the curve is

(4)

The curvature and tangential angle are given by

(5)

for . There does not seem to be a simple closed form for the arc length of the curve, but its numerical value is approximately given by .