how can I graph a bicorn given only its equation?

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

[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 $x$.

Answer 4

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

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

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In Mathematica:

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

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