Bicorn

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For the hat, see Bicorne.
For the mythical beast, see Bicorn (legendary creature).
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[edit]

In 1864, James Joseph Sylvester studied the curve

y^4-xy^3-8xy^2+36x^2y+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[edit]

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

See also[edit]

References[edit]