Degenerate Conic - Applications

Applications

Degenerate conics, as with degenerate algebraic varieties generally, arise as limits of non-degenerate conics, and are important in compactification of moduli spaces of curves.

For example, the pencil of curves (1-dimensional linear system of conics) defined by is non-degenerate for but is degenerate for concretely, it is an ellipse for two parallel lines for and a hyperbola with – throughout, one axis has length 2 and the other has length which is infinity for

Such families arise naturally – given four points in general linear position (no three on a line), there is a pencil of conics through them (five points determine a conic, four points leave one parameter free), of which three are degenerate, each consisting of a pair of lines, corresponding to the ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient).

For example, given the four points the pencil of conics through them can be parameterized as yielding the following pencil; in all cases the center is at the origin:

• hyperbolae opening left and right;
• the parallel vertical lines
• ellipses with a vertical major axis;
• a circle (with radius );
• ellipses with a horizontal major axis;
• the parallel horizontal lines
• hyperbolae opening up and down,
• the diagonal lines

(dividing by and taking the limit as yields )

• This then loops around to since pencils are a projective line.

Note that this parametrization has a symmetry, where inverting the sign of a reverses x and y. In the terminology of (Levy 1964), this is a Type I linear system of conics, and is animated in the linked video.

A striking application of such a family is in (Faucette 1996) which gives a geometric solution to a quartic equation by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the resolvent cubic.

Pappus's hexagon theorem is the special case of Pascal's theorem, when a conic degenerates to two lines.