Bicircular quartic curves

2 January 2009 at 5:33 pm 1 comment

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).

Following the definition, one can give the general form of bicircular quartics : setting x=1, the equation should vanish multiply at the points I = (i,0) and J = (-i,0), and thus have the form

(y^2 + 1)^2 + z P(y,z) = 0

where P(y,z) = (y^2+1)(2a+2by) + zQ(y,z) to ensure vanishing of the derivatives at I and J. If Q vanishes at (\pm i, 0), I and J are actually cusps. We obtain the general equation

(x^2 + y^2)^2 + (x^2 + y^2)(2ax+2by)z + z^2 Q(x,y,z) = 0

or
(x^2 + y^2 + axz + byz)^2 + z^2 Q(x,y,z) = 0

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 2xy=1, and thus has equation (x^2+y^2)^2 = 2xy : 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.

Entry filed under: curves, english, geometry. Tags: , , , , .

Inversion and n-circular curves Intégrales elliptiques et moyenne arithmético-géométrique

1 Comment Add your own

  • 1. Pierre  |  5 January 2009 at 4:32 pm

    C’était quand même mieux quand on avait droit en sus à une version française🙂

    Reply

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