Bicorn

For the hat, see Bicorne.

For the mythical beast, see Bicorn and Chichevache.

Bicorn

In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation

$y^{2}(a^{2}-x^{2})=(x^{2}+2ay-a^{2})^{2}.$

It has two cusps and is symmetric about the y-axis.

History

In 1864, James Joseph Sylvester studied the curve

$y^{4}-xy^{3}-8xy^{2}+36x^{2}y+16x^{2}-27x^{3}=0$

in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867.

Properties

A transformed bicorn with a = 1

The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at x=0, z=0 . If we move x=0 and z=0 to the origin substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve, we obtain

$(x^{2}-2az+a^{2}z^{2})^{2}=x^{2}+a^{2}z^{2}.\,$

This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i and z=1.

The parametric equations of a bicorn curve are:

$x=a\sin(\theta )$ and $y=a{\frac {\cos ^{2}(\theta )\left(2+\cos(\theta )\right)}{3+\sin ^{2}(\theta )}}$ with $-\pi \leq \theta \leq \pi$

See also

• List of curves

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 147–149. ISBN 0-486-60288-5.
• "Bicorn" at The MacTutor History of Mathematics archive
• Weisstein, Eric W. "Bicorn". MathWorld.
• "Bicorne" at "mathcurve"
• The Collected Mathematical Papers of James Joseph Sylvester. Vol. II Cambridge (1908) p. 468 (online)