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Bicorn

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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

Contents

y 2 ( a 2 − x 2 ) = ( x 2 + 2 a y − 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 − x y 3 − 8 x y 2 + 36 x 2 y + 16 x 2 − 27 x 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

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 − 2 a z + 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 ⁡ ( θ ) and y = cos 2 ⁡ ( θ ) ( 2 + cos ⁡ ( θ ) ) 3 + sin 2 ⁡ ( θ ) with − π ≤ θ ≤ π

References

Bicorn Wikipedia
(Text) CC BY-SA