geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the bicorn, also known as a cocked hat curve due to its resemblance to a

bicorne The bicorne or bicorn (two-cornered) is a historical form of hat widely adopted in the 1790s as an item of uniform by European and American army and naval officers. Most generals and staff officers of the Napoleonic period wore bicornes, whic ...

, is a

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...

quartic curve defined by the equation

y^2 \left(a^2 - x^2\right) = \left(x^2 + 2ay - a^2\right)^2.

It has two

cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...

s and is symmetric about the y-axis.

History

In 1864,

James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ro ...

studied the curve

y^4 - xy^3 - 8xy^2 + 36x^2y+ 16x^2 -27x^3 = 0

in connection with the classification of

quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a q ...

s; he named the curve a bicorn because it has two cusps. This curve was further studied by

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...

in 1867.

Properties

The bicorn is a

plane algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

of degree four and

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

zero. It has two cusp singularities in the real plane, and a double point in the

complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...

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

\left(x^2 - 2az + a^2 z^2\right)^2 = x^2 + a^2 z^2.

This curve, a

limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...

, has an ordinary double point at the origin, and two nodes in the complex plane, at and . The parametric equations of a bicorn curve are

x = a \sin(\theta)

and

y = a \frac

with

-\pi\le\theta\le\pi

References

External links

* {{MathWorld, title=Bicorn, urlname=Bicorn Plane curves Algebraic curves

History

Properties

See also

References

External links