= Cissoid of Diocles
{title2=$y^2(2a-x)=x^3$}
{wiki}
For $a>0$, this <algebraic curve> can be constructed using a <circle> of <radius> $a$ centred at $(a,0)$ and its <tangent line> $x=2a$. On a ray from the origin, subtract the distance to its nonzero circle intersection from the distance to the line. The resulting radial distance is $r=2a\sin^2\theta/\cos\theta$, for $-\pi/2<\theta<\pi/2$. Thus a <parametric curve> for it is
$$
x=\frac{2at^2}{1+t^2},\qquad y=\frac{2at^3}{1+t^2},\qquad t\in\mathbb R.
$$
Eliminating $t$ gives the displayed cubic equation. It has a <cusp> at the origin, horizontal tangent direction there, and vertical <asymptote> $x=2a$. Its name honours <Diocles>.
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