Bicorn

 ( Plane Curves ) 

In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a Rational curve quartic curve defined by the equation

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

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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 ; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867.

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Properties The bicorn is a algebraic curve of degree four and geometric 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 number 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 $z=1$.

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

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See also

• List of curves

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External links

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