Bicorn

For the hat, see Bicorne.

For the mythical beast, see Bicorn (monster).

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

y²(a² − x²) = (x² + 2ay − a²)².

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

History

In 1864, James Joseph Sylvester studied the curve

y⁴ − xy³ − 8xy² + 36x²y + 16x² − 27x³ = 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

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

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

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.

A transformed bicorn with a = 1

.

The parametric equations of a bicorn curve are:

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

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" from MathWorld.
• "Bicorne" at Encyclopédie des Formes Mathématiques Remarquables
• The Collected Mathematical Papers of James Joseph Sylvester. Vol. II Cambridge (1908) p. 468 (online)