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what are the parts or the variables present in the bicorn equation?

4 Answers 4

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According to the McGraw-Hill Dictionary of Scientific & Technical Terms, the bicorn curve is given by the solution set of $(x^2 + 2ay - a^2)^2 = y^2(a^2 - x^2)$, where $a$ is an arbitrary constant. The reference can be found here.

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    I would say that$a$was a scale constant and that $x$ and $y$ were the variables. The equation is a quadratic for $y$, so it is easy to draw, with two values of $y$ for each $x \in (-a,a)$. You can find more in [Wikipedia](http://en.wikipedia.org/wiki/Bicorn)2011-03-22
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In geometry, the bicorn, also known as a cocked hat curve or bicorne is a rational quartic curve which has two cusps. 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.

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[Copied from the duplicate question]

Is the equation $y^2(a^2-x^2)=(x^2+2ay-a^2)^2$ (as suggested here) what you are trying to draw?

If so, it is a quadratic equation for $y$ which you can solve to something like

$y = \frac{(a^2-x^2)\left( 2 a \pm \sqrt{ a^2 -x^2} \right)}{3 a^2 + x^2}$

and then draw. You need $a^2 -x^2$ to be non-negative, and that gives you the range of $x$.

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I find it more convenient to use the parametric equations for the bicorn:

$\begin{align*}x&=a\cos\,t\\y&=a\frac{\sin^2 t}{2+\sin\,t}\end{align*}$


In Mathematica:

ParametricPlot[{Cos[t], Sin[t]^2/(2 + Sin[t])}, {t, 0, 2 Pi},     Axes -> None, Frame -> True] 

bicorn

This checks that the equations are right:

y^2 (a^2 - x^2) == (x^2 + 2 a y - a^2)^2 /.    Thread[{x, y} -> {a Cos[t], a Sin[t]^2/(2 + Sin[t])}] // FullSimplify True