A family of 4th degree curves

A family of 4th degree rational curves

The curves in this family are rational curves of degree 4. All but finitely many of the curves in this family have 3 ordinary double points (or nodes). There is one parameter value where the nodes coalesce to form an ordinary triple point [with 3 distinct tangent directions], and there is a parameter value where the nodes change into cusps.

The equation of a curvein this family is:

b²(x²+y²)² - 2a²b(x³-3xy²) -(b²-a²)(a²+2b²)(x²+y²) + (b²-a²)³ = 0,

where a and b are real numbers and a + b=1. Thus, we can eliminate a from the equation, and we have a 1-parameter family. The curves shown in the picture are typical of the values of b between 1/3 and 1. The following parameters are actually shown:

b= 8/15
This curve has 3 singular points; the singular points are nodes. In fact, this is true for most of the curves in the family.
b= 1/2
The origin is the only singular point of this curve; it is an ordinary triple point. Intuitively, we can think of the 3 nodes of a generic curve as coming together to form the triple point.
b= 2/5
This curve has 3 nodes. So, it is another generic member of the family.
b= 1/3
This curve is also exceptional; it has 3 singular points but they are cusps rather than nodes. Visually, it appears that as b approaches 1/3, each of the 3 loops shrinks to a point, and the two tangent lines at each node come closer to coinciding.

More about the equation The equation can be written as follows in polar coordinates:

b²r⁴ - 2a²br³cos(3θ) - (b²-a²)(a²+2b²)r⁴ + (b²-a²)³ = 0.

The fact that the equation can be expressed in terms of 3θ explains the rotational symmetry.
Note also that if a = b or a = - b, then the equation in polar coordinates becomes br = 2a²cos(3θ).

Determination of the genus For a nonsingular plane curve of degree d, the genus is (d-1)(d-2)/2. So, a nonsingular 4^th degree curve has genus 3. A node reduces the genus by 1. Since a generic curve in this family has 3 nodes, its genus is 3 - 3 = 0; thus it must be rational. The exceptional curve with the triple point (corresponding to b= 0) is also rational. This is shown explicitly by the sketch of its projection from the origin.

About the drawingThe curves were plotted by means of the "implicitplot" command in the plots package of Maple, and then combined by means of the "display" command. The grid size was about 150 by 150 -- or more specifically 151 by 151 or 145 by 145 for the various curves to increase the likelihood of actually plotting the singular points. Nonetheless, there still were a few small gaps; therefore the .gif image has been lightly re-touched.

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