Degenerate conic

Main article: Conic section

In mathematics, a degenerate conic is a conic (degree-2 plane curve, the zeros of a degree-2 polynomial equation, a quadratic) that fails to be an irreducible curve. This can happen in two ways: either it is a reducible variety, meaning that its defining quadratic factors as the product of two linear polynomials (degree 1), or the polynomial is irreducible but does not define a curve, but instead a lower-dimension variety (a point or the empty set); this latter can only occur over a field that is not algebraically closed, such as the real numbers.

Examples

As an example of the first failure, reducibility, x² − y² = 1 is not degenerate (it defines a hyperbola), but x² − y² = 0 is degenerate because it is reducible – it factors as x² − y² = (x − y)(x + y), and corresponds to two intersecting lines.

As an example of the second failure, not enough points (over the field of definition), over the real numbers x² + y² = 1 is not degenerate (it defines a circle) but x² + y² = 0 is degenerate – it defines a point, (0,0), not a curve, and x² + y² = − 1 is likewise degenerate, defining the empty set. Note that over the complex numbers x² + y² factors as (x + iy)(x − iy) and is degenerate because reducible, while x² + y² = − 1 defines a non-degenerate curve, an ellipse/hyperbola (these are not distinct over the complex numbers, because there is no sense of positive or negative).

Classification

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

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, such as x² − y² = (x + y)(x − y) = 0 – a 4-dimensional space
• Two parallel lines, such as x² − 1 = (x + 1)(x − 1) = 0 – a 3-dimensional space
• A single doubled line (multiplicity 2), such as x² = 0 – a 2-dimensional space

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

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 x² + y² = 0.
• No points, such as x² + y² = − 1 – an imaginary ellipse.

Discriminant

Just as non-degenerate real conics can be classified by the discriminant of their imaginary part, considered as a quadratic form in (x,y), (the determinant of the matrix of the associated symmetric form), a conic is degenerate if and only if the discriminant of the homogeneous quadratic form in (x,y,z) is zero,^[1] where the affine equation

Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F

(factors of 2 for cross terms) is homogenized to

Ax² + 2Bxy + Cy² + 2Dxz + 2Eyz + Fz²;

the discriminant in this sense is then the determinant of the matrix:

A B D
B C E
D E F

Recall that the discriminant for the elliptic/parabolic/hyperbolic is the determinant of the matrix:

A B
B C

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 x² + ay² = 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 videos
	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)x² + (1 − a)y² = 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 x² − y² = 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

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 x² − y² = a², or to two parallel lines: x² − a²y² = 1, or to double line: x² − a²y² = a²,
• parabolae can degenerate to two parallel lines: x² − ay − 1 or a double line x² − ay, but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
• ellipses can degenerate to two parallel lines: x² + a²y² − 1 or a double line x² + a²y² − a², but, because they have conjugate complex points at infinity which become a double point on degeneration, cannot degenerate to two intersecting lines.

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 x² − ay² − 1, or to a double line by rotating into each other about a point, as in x² − ay².
• Two parallel lines can degenerate to a double line by moving into each other, as in x² − a², but cannot degenerate to non-parallel lines.
• A double line cannot degenerate to the other types.

Points to define

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.

Degenerate ellipse with semiminor axis of zero

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.

Notes

1. ^ A simpler parametrization is given by ax² + (1 − a)y² = 1, which are the affine combinations of the equations x² = 1 and y² = 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

1. ^ (Lasley, Jr. 1957)

• Coffman, Adam, Linear Systems of Conics, http://www.ipfw.edu/math/Coffman/pov/lsoc.html 
• Faucette, William Mark (January 1996), "A Geometric Interpretation of the Solution of the General Quartic Polynomial", The American Mathematical Monthly 103 (1): 51–57, JSTOR 2975214, alternative download 
• Lasley, Jr., J. W. (May 1957), "On Degenerate Conics", The American Mathematical Monthly (Mathematical Association of America) 64 (5): 362–364, JSTOR 2309606 
• Levy, Harry (1964), Projective and related geometries, New York: The Macmillan Co., pp. x+405 
• Milne, J. J. (January 1926), "Note on Degenerate Conics", The Mathematical Gazette (The Mathematical Association) 13 (180): 7–9, JSTOR 3602237 
• "7.2 The General Quadratic Equation", CRC Standard Mathematical Tables and Formulas (30th ed.), http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node28.html