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

The conic section with equation <math>x^2 + y^2 = 0</math> 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, <math>(0,0)</math>, not a curve.

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

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

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

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 point]]s (if it has any at all). This can occur in two ways:
* A single double point, such as <math>x^2 + y^2 = 0.</math>
* No points, such as <math>x^2+y^2 = -1</math> – an '''imaginary ellipse'''.

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

:<math>\begin{bmatrix} A & B \\ B & C \\ \end{bmatrix}, </math>

the matrix of the quadratic form in <math>(x,y)</math>.

Analogously, a conic can be classified as non-degenerate or degenerate according to the discriminant of the ''homogeneous'' quadratic form in <math>(x,y,z)</math>.<ref>{{Harv|Lasley, Jr.|1957}}</ref> Here the affine form is homogenized to
:<math>Ax^2 + 2Bxy + Cy^2 +2Dxz + 2Eyz + Fz^2;</math>
the discriminant of this form is the determinant of the matrix:
:<math>\begin{bmatrix} A & B & D \\ B & C & E \\ D & E & F \\ \end{bmatrix}.</math>

The conic is degenerate if and only of 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==

Conics, also known as conic sections to emphasize their three-dimensional geometry, arise as the intersection of a [[plane (geometry)|plane]] with a [[cone]]. Degeneracy occurs when the plane contains the [[Apex (geometry)|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 ==
Degenerate conics, as with degenerate [[algebraic varieties]] generally, arise as limits of non-degenerate conics, and are important in [[Compactification (mathematics)|compactification]] of [[moduli of algebraic curves|moduli spaces of curves]].

For example, the [[Pencil (mathematics)|pencil]] of curves (1-dimensional [[linear system of conics]]) defined by <math>x^2 + ay^2 = 1</math> is non-degenerate for <math>a\neq 0</math> but is degenerate for <math>a=0;</math> concretely, it is an ellipse for <math>a>0,</math> two parallel lines for <math>a=0,</math> and a hyperbola with <math>a<0</math> – throughout, one axis has length 2 and the other has length <math>1/\sqrt{|a|},</math> which is infinity for <math>a=0.</math>

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

{{external media | video1 = [http://www.ipfw.edu/math/Coffman/pov/conic1.gif Type I] linear system, {{Harv|Coffman}}.}}
For example, given the four points <math>(\pm 1, \pm 1),</math> the pencil of conics through them can be parameterized as <math>(1+a)x^2+(1-a)y^2=2,</math> yielding the following pencil; in all cases the center is at the origin:<ref group="note">A simpler parametrization is given by <math>ax^2+(1-a)y^2=1,</math> which are the [[affine combination]]s of the equations <math>x^2=1</math> and <math>y^2=1,</math> corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of <math>0,1,\infty.</math></ref>
* <math>a>1:</math> hyperbolae opening left and right;
* <math>a=1:</math> the parallel vertical lines <math>x=-1, x=1;</math>
* <math>0 < a < 1:</math> ellipses with a vertical major axis;
* <math>a=0:</math> a circle (with radius <math>\sqrt{2}</math>);
* <math>-1 < a < 0:</math> ellipses with a horizontal major axis;
* <math>a=-1:</math> the parallel horizontal lines <math>y=-1, y=1;</math>
* <math>a<-1:</math> hyperbolae opening up and down,
* <math>a=\infty:</math> the diagonal lines <math>y=x, y=-x;</math>
:(dividing by <math>a</math> and taking the limit as <math>a \to \infty</math> yields <math>x^2-y^2=0</math>)
* This then loops around to <math>a>1,</math> 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 {{Harv|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 {{Harv|Faucette|1996}} which gives a [[Quartic formula#Algebraic geometry|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 <math>x^2-y^2=a^2,</math> or to two parallel lines: <math>x^2-a^2y^2=1,</math> or to double line: <math>x^2-a^2y^2=a^2,</math>
* Parabolae can degenerate to two parallel lines: <math>x^2-ay-1=0</math> or a double line <math>x^2-ay=0,</math> but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
* Ellipses can degenerate to two parallel lines: <math>x^2+a^2y^2-1=0</math> or a double line <math>x^2+a^2y^2-a^2=0,</math> 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 <math>x^2-ay^2-1=0,</math> or to a double line by rotating into each other about a point, as in <math>x^2-ay^2=0.</math>
* Two parallel lines can degenerate to a double line by moving into each other, as in <math>x^2-a^2=0,</math> 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 <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2} = 1</math>, 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 [[Focus (geometry)|foci]] at the endpoints. As an [[orbit]], this is a [[Elliptic orbit#Radial elliptic trajectory|radial elliptic trajectory]].

== 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.

== Notes ==
{{reflist|group = note}}

== References ==
{{reflist}}
{{refbegin}}
* {{ Citation | first = Adam | last = Coffman | url = http://users.ipfw.edu/CoffmanA/pov/lsoc.html | title = Linear Systems of Conics }}
* {{ Citation | title = A Geometric Interpretation of the Solution of the General Quartic Polynomial | first = William Mark | last = Faucette | journal = [[The American Mathematical Monthly]] | volume = 103 | number = 1 |date=January 1996 | pages = 51–57 | jstor = 2975214 | id = {{citeseerx|10.1.1.111.5574}} }}
* {{ Citation | title = On Degenerate Conics | first = J. W. | last = Lasley, Jr. | journal = [[The American Mathematical Monthly]] | volume = 64 | number = 5 |date=May 1957 | pages = 362–364 | publisher = [[Mathematical Association of America]] | jstor = 2309606 }}
* {{ Citation | last = Levy | first = Harry | title = Projective and related geometries | publisher = The Macmillan Co. | location = New York | year = 1964 | pages = x+405 }}
* {{ Citation | title = Note on Degenerate Conics | first = J. J. | last = Milne | journal = The Mathematical Gazette | volume = 13 | number = 180 |date=January 1926 | pages = 7–9 | publisher = The Mathematical Association | jstor = 3602237 }}
* {{ Citation | title = CRC Standard Mathematical Tables and Formulas | edition = 30th | chapter = 7.2 The General Quadratic Equation | chapterurl = http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node28.html }}
{{refend}}

[[Category:Conic sections]]

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