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] It has two cusps and is symmetric about the y-axis.[2]
History
In 1864, James Joseph Sylvester studied the curve 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 . If we move and to the origin and perform an imaginary rotation on by substituting for and for in the bicorn curve, we obtain This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at and .[4]
The parametric equations of a bicorn curve are with
See also
References
- ^ Lawrence, J. Dennis (1972). A catalog of special plane curves. Dover Publications. pp. 147–149. ISBN 0-486-60288-5.
- ^ "Bicorn". mathcurve.
- ^ The Collected Mathematical Papers of James Joseph Sylvester. Vol. II. Cambridge: Cambridge University press. 1908. p. 468.
- ^ "Bicorn". The MacTutor History of Mathematics.
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