## Posts tagged ‘quartic’

### 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 $h^1(\mathcal O_X) = 3$, 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 $MA \cdot MB = k^2$ where A and B are fixed foci, while Descartes define Cartesian ovals by the equation $a \cdot MA + b \cdot MB = k$ (these equations are equivalently to algebraic quartic equations, using appropriate squarings).