Bicircular quartic curves

Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. We are usually interested in real curves, so the type of singulaity is the same at I or J. Salmon in his Treatise on Higher Plane Curves and Basset in his Elementary Treatise on Cubic and Quartic curves deal in detail with these curves.

A plane quartic curve has arithmetic genus , but since bicircular quartics have at least two double points, they have geometric genus 0 or 1 (genus of the desingularized curve).
Curves having geometric genus 0 are called rational curves (formerly known as unicursal curves), and admit a parameterization by rational functions of one variable.

Families of bicircular curves were defined by Cassini and Descartes by metric properties : Cassini ovals are defined by the equation where A and B are fixed foci, while Descartes define Cartesian ovals by the equation (these equations are equivalently to algebraic quartic equations, using appropriate squarings).

Inversion and n-circular curves

Inversion is a transformation of the Euclidean plane closely related to circles. It is seldom taught in high schools (at least in France), but it has special interest when dealing with circles, since it maps circles and lines to circles or lines. Inversion is defined with respect to a center O and a radius ρ (in French, the inversion is said to have pôle O and puissance (power) ρ²).