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).
Following the definition, one can give the general form of bicircular quartics : setting , the equation should vanish multiply at the points and , and thus have the form
where to ensure vanishing of the derivatives at I and J. If Q vanishes at , I and J are actually cusps. We obtain the general equation
where Q is a quadratic polynomial. A bicircular quartic is thus obtained as an element of a pencil containing the double of a circle and a divisor containing twice the line at infinity.
Another property of bicircular quartics is that for any point O in the plane, the product of algebraic distances to the four intersection points on a line through O does not depend on the choice of the line.
Bernoulli’s lemniscate is an example of rational bicircular quartic : it can be written as the inverse of the equilateral hyperbola , and thus has equation : it has two nodes at the circular points and one at the origin, as well as nice properties. General rational bicircular quartics have a lot of different characterization : for example, any such curve is obtained from a conic by inversion. The center of inversion become a singular point of the image curve.
Many “classical” quartic curves are bicircular : the inversion of a conic whith respect to one of its foci is called a Pascal’s limaçon, and includes the cardiod (inversion of a parabola). More generally, circular curves appear naturally when equations involve distances, or when studying loci of points involving circles.