 Cissoid of Diocles

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 isEliminating $t$ 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.

The Cissoid of Diocles is a notable mathematical curve from ancient Greek geometry, named after the Greek mathematician Diocles, who studied it around the 2nd century BCE. It is defined in the context of a specific geometrical construction involving a circle and lines, and it has applications in the creation of certain types of solutions for cubic equations.