Degenerate conic

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Main article: Conic section

In mathematics, a degenerate conic is a conic (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) that fails to be an irreducible curve. This can happen in two ways: either it is a reducible variety, meaning that its defining quadratic form is factorable as the product of two linear polynomials, or the polynomial is irreducible but defines not a curve but instead a lower-dimension variety (a point or the empty set); the latter can only occur over a field that is not algebraically closed, such as the real numbers.

In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (actually two coinciding lines), a single point, or the null set (no points).

Examples[edit]

The conic section with equation x^2-y^2 = 0 is an example of the first failure, reducibility. This conic section is degenerate because it is reducible. The equation can be written as (x-y)(x+y)= 0, and corresponds to two intersecting lines or an "X".

The conic section with equation x^2 + y^2 = 0 is an example of the second failure, not enough points (over the field of definition), over the real numbers. This conic section is degenerate because it defines only one point, (0,0), not a curve.

The conic section with equation x^2+ y^2 = -1 is likewise degenerate because it defines the empty set.

Over the field of complex numbers, the conic section with equation x^2 + y^2 = 0 factors as (x+iy)(x-iy)=0 and is degenerate because it is reducible.

Classification[edit]

Over the complex projective plane there are only two types of degenerate conics – two different lines, which necessarily intersect in one point, or one double line.

Over the real affine plane the situation is more complicated.

Reducible[edit]

Reducible conics – those whose equation factors – consist of two lines in the plane. There are three possible configurations of these, according to how they intersect. These form a 4-dimensional space (each line has two parameters, namely a slope and a position, as is slope-intercept form), with special intersections as lower-dimensional sub-varieties.

  • Two intersecting lines, a 4-dimensional space, such as x^2-y^2 = 0 \Leftrightarrow (x+y)(x-y) = 0
  • Two parallel lines,a 3-dimensional space, such as x^2-1 = 0 \Leftrightarrow (x+1)(x-1) = 0
  • A single doubled line (multiplicity 2), a 2-dimensional space, such as x^2 = 0

In terms of the points at infinity, two intersecting lines have 2 distinct points at infinity, while two parallel lines intersect at 1 point at infinity (hence intersect the line at infinity in a double point), and a single double line also intersects the line at infinity in a double point.

Not enough points[edit]

Over a non-algebraically closed field such as the real numbers, a conic may also be degenerate because it does not have enough real points (if it has any at all). This can occur in two ways:

  • A single double point, such as x^2 + y^2 = 0.
  • No points, such as x^2+y^2 = -1 – an imaginary ellipse.

Discriminant[edit]

Non-degenerate real conics can be classified as ellipses, parabolas, or hyperbolas by the discriminant of the non-homogeneous form Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F, which is the determinant of the matrix

\begin{bmatrix} A & B \\ B & C \\ \end{bmatrix},

the matrix of the quadratic form in (x,y).

Analogously, a conic can be classified as non-degenerate or degenerate according to the discriminant of the homogeneous quadratic form in (x,y,z).[1] Here the affine form is homogenized to

Ax^2 + 2Bxy + Cy^2 +2Dxz + 2Eyz + Fz^2;

the discriminant of this form is the determinant of the matrix:

\begin{bmatrix} A & B & D \\ B & C & E \\ D & E & F \\ \end{bmatrix}.

The conic is degenerate if and only if the determinant of this matrix equals zero.

See Matrix representation of conic sections#Classification for determination of the specific type of conic and the specific type of degeneracy based on the parameters of the conic.

Relation to intersection of a plane and a cone[edit]

Conics, also known as conic sections to emphasize their three-dimensional geometry, arise as the intersection of a plane with a cone. Degeneracy occurs when the plane contains the apex of the cone or when the cone degenerates to a cylinder and the plane is parallel to the axis of the cylinder. See Conic section#Degenerate cases for details.

Applications[edit]

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 x^2 + ay^2 = 1 is non-degenerate for a\neq 0 but is degenerate for a=0; concretely, it is an ellipse for a>0, two parallel lines for a=0, and a hyperbola with a<0 – throughout, one axis has length 2 and the other has length 1/\sqrt{|a|}, which is infinity for a=0.

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 \textstyle{\binom{4}{2,2}=3} ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient).

External video
Type I linear system, (Coffman).

For example, given the four points (\pm 1, \pm 1), the pencil of conics through them can be parameterized as (1+a)x^2+(1-a)y^2=2, yielding the following pencil; in all cases the center is at the origin:[note 1]

  • a>1: hyperbolae opening left and right;
  • a=1: the parallel vertical lines x=-1, x=1;
  • 0 < a < 1: ellipses with a vertical major axis;
  • a=0: a circle (with radius \sqrt{2});
  • -1 < a < 0: ellipses with a horizontal major axis;
  • a=-1: the parallel horizontal lines y=-1, y=1;
  • a<-1: hyperbolae opening up and down,
  • a=\infty: the diagonal lines y=x, y=-x;
(dividing by a and taking the limit as a \to \infty yields x^2-y^2=0)
  • This then loops around to a>1, 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.

Degeneration[edit]

In the complex projective plane, all conics are equivalent, and can degenerate to either two different lines or one double line.

In the real affine plane:

  • Hyperbolae can degenerate to two intersecting lines (the asymptotes), as in x^2-y^2=a^2, or to two parallel lines: x^2-a^2y^2=1, or to double line: x^2-a^2y^2=a^2,
  • Parabolae can degenerate to two parallel lines: x^2-ay-1=0 or a double line x^2-ay=0, but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
  • Ellipses can degenerate to two parallel lines: x^2+a^2y^2-1=0 or a double line x^2+a^2y^2-a^2=0, but, because they have conjugate complex points at infinity which become a double point on degeneration, cannot degenerate to two intersecting lines.Complex plane do represent a conic section of imaginary fused state between an ellipse and a parabola.

Degenerate conics can degenerate further to more special degenerate conics, as indicated by the dimensions of the spaces and points at infinity.

  • Two intersecting lines can degenerate to two parallel lines, by rotating until parallel, as in x^2-ay^2-1=0, or to a double line by rotating into each other about a point, as in x^2-ay^2=0.
  • Two parallel lines can degenerate to a double line by moving into each other, as in x^2-a^2=0, but cannot degenerate to non-parallel lines.
  • A double line cannot degenerate to the other types.
  • Another type of degeneration occurs when an ellipse, rotated and translated to its simplest form \tfrac{x^2}{a^2}+\tfrac{y^2}{b^2} = 1, has its semiminor axis b go to zero and thus has its eccentricity go to one. The result is a line segment (degenerate because the ellipse is not differentiable at the endpoints) with its foci at the endpoints. As an orbit, this is a radial elliptic trajectory.

Points to define[edit]

A general conic is defined by five points: given five points in general position, there is a unique conic passing through them. If three of these points lie on a line, then the conic is reducible, and may or may not be unique. If no four points are collinear, then five points define a unique conic (degenerate if three points are collinear, but the other two points determine the unique other line). If four points are collinear, however, then there is not a unique conic passing through them – one line passing through the four points, and the remaining line passes through the other point, but the angle is undefined, leaving 1 parameter free. If all five points are collinear, then the remaining line is free, which leaves 2 parameters free.

Given four points in general linear position (no three collinear; in particular, no two coincident), there are exactly three pairs of lines (degenerate conics) passing through them, which will in general be intersecting, unless the points form a trapezoid (one pair is parallel) or a parallelogram (two pairs are parallel).

Given three points, if they are non-collinear, there are three pairs of parallel lines passing through them – choose two to define one line, and the third for the parallel line to pass through, by the parallel postulate.

Given two distinct points, there is a unique double line through them.

Notes[edit]

  1. ^ A simpler parametrization is given by ax^2+(1-a)y^2=1, which are the affine combinations of the equations x^2=1 and y^2=1, corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of 0,1,\infty.

References[edit]