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Bicorn

Bicorn.

The bicorn, also called the cocked hat curve, is the name of a collection of quartic curves studied by James Sylvester in 1864. The same curves were studied by Arthur Cayley in 1867.

The equation of the bicorn in Cartesian coordinates is

y2(a2−x2)=(x2+2ay−a2)2y^2\left(a^2-x^2\right)=\left(x^2+2ay-a^2\right)^2

The parametric equations of the bicorn can be given by

x=asin⁡t,y=a(2+cos⁡t)cos2⁡t3+sin2⁡tx=a\sin t,\,y=a\frac{\left(2+\cos t \right)\cos^2t}{3+\sin^2 t}

In polar coordinates, the equation of the bicorn is given by

(rsin⁡θ)2(a2−(rcos⁡θ)2)=((rcos⁡θ)2+2a(rsin⁡θ)−a2)2\left(r\sin \theta\right)^2\left(a^2-\left(r\cos \theta \right)^2\right)=\left(\left(r\cos \theta \right)^2+2a\left(r \sin \theta \right)-a^2\right)^2

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