Finding Degenerate Points in Conics with Projective Geometry

Basic explanation

Conic sections - circles, ellipses, parabolas, hyperbolas - arise as plane sections of a double cone, or as the locus of points with fixed eccentricity relative to a focus and directrix. For special coefficients the quadratic collapses: for example $x^2 - y^2 = 0$ factors as $(x-y)(x+y)=0$ and describes two intersecting lines. Such reducible curves are degenerate conics; singular points on them are places where the usual smooth-curve picture fails.

This paper was an individual AMP research project at the University of Liverpool Mathematics School. The aim was to explain, from modest prerequisites, how to recognise degeneracy and locate singular points using projective geometry.

Degenerate cases

Geometrically, degeneracy appears when the cutting plane passes through the apex of the double cone: the section can become a point, a line, or a pair of intersecting lines. Algebraically there is also a “lost” case of two parallel lines (a hyperbola with eccentricity tending to infinity) that cannot be realised by a planar cut of the cone. Singular points are especially meaningful for the point and intersecting-line cases; a lone line has no isolated singular point in the same sense.

Formally, a degenerate conic is a degree-two plane curve that factors over an algebraically closed field into linear factors - so the curve is reducible rather than an irreducible ellipse, parabola, or hyperbola.

Projective method

Detecting factorability by hand can be awkward. The method developed in the paper uses the observation that singular points have no well-defined gradient in the usual affine picture, then lifts the problem to a surface where critical points are accessible.

Start from

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.$

Homogenise by viewing the conic in the plane $z = 1$ and forming the cone of rays from the origin through that curve:

$Ax^2 + Bxy + Cy^2 + Dxz + Eyz + Fz^2 = 0.$

The three partial derivatives with respect to $x$, $y$, and $z$ yield a linear system. A vanishing determinant of that system signals degeneracy; solving the system and returning to $z = 1$ recovers the singular point in the affine plane.

Why it matters

Beyond classification, singular loci appear in optics (lens profiles defined by conics), celestial mechanics, and links between poles of complex functions and geometric singularities. The paper’s contribution is not a new theorem for specialists but a self-contained route - definitions, examples, homogenisation, and the determinant test - that makes the projective viewpoint usable at sixth-form level while staying mathematically honest.