Bicorn

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.{{cite book | title = The Collected Mathematical Papers of James Joseph Sylvester | volume = II | location = Cambridge | year = 1908 | page = 468 | url = https://archive.org/details/collectedmathem01sylvrich | publisher = Cambridge University press}}

File:Bicorn-inf.jpg

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 $x=\; \backslash pm\; i$ and $z=1$.{{cite web | url = http://www-history.mcs.st-andrews.ac.uk/history/Curves/Bicorn.html | title= Bicorn | work = The MacTutor History of Mathematics}}

The parametric equations of a bicorn curve are

$$\backslash begin\{align\}$$

x &= a \sin\theta \\

y &= a \, \frac{(2 + \cos\theta) \cos^2\theta}{3 + \sin^2\theta}

\end{align}

with $-\backslash pi\; \backslash le\; \backslash theta\; \backslash le\; \backslash pi.$

• List of curves

{{reflist}}

• {{MathWorld|title=Bicorn|urlname=Bicorn}}

Category:Plane curves

Category:Quartic curves