Bicorn Explained

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^[1] $$y^2\; \backslash left(a^2\; -\; x^2\backslash right)\; =\; \backslash left(x^2\; +\; 2ay\; -\; a^2\backslash right)^2.$$It has two cusps and is symmetric about the y-axis.^[2]

History

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.^[3]

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 and perform an imaginary rotation on x by substituting ix/z for x and 1 /z for y in the bicorn curve, we obtain$$\backslash left(x^2\; -\; 2az\; +\; a^2\; z^2\backslash right)^2\; =\; x^2\; +\; a^2\; z^2.$$This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = \pm i and z =1 .^[4]

The parametric equations of a bicorn curve are with

- \pi \le \theta \le \pi.

See also

• List of curves

Notes and References

1. Book: Lawrence, J. Dennis . A catalog of special plane curves . Dover Publications . 1972 . 0-486-60288-5 . 147–149 . registration .
2. Web site: Bicorn . mathcurve.
3. Book: The Collected Mathematical Papers of James Joseph Sylvester . II . Cambridge . 1908 . 468 . Cambridge University press.
4. Web site: Bicorn . The MacTutor History of Mathematics.