Bicorn

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Bicorn
Bicorn

The bicorn is also known as a cocked hat curve, due to its resemblance to a bicorne. It is a curve with equation

y^2(a^2-x^2)=(x^2+2ay-a^2)^2 \,

with two cusps, symmetrical around the y-axis.

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
A transformed bicorn with a = 1
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