${\displaystyle x^2-y^2=0}$

${\displaystyle x^2-1=0}$

${\displaystyle x^2=0}$

${\displaystyle x^2+y^2=0}$

Ingeometry, adegenerate conic is aconic (a second-degreeplane curve, defined by apolynomial equation of degree two) that fails to be anirreducible curve. This means that the defining equation is factorable over thecomplex numbers (or more generally over analgebraically closed field) as the product of two linear polynomials.

Using the alternative definition of the conic as the intersection inthree-dimensional space of aplane and a doublecone, a conic is degenerate if the plane goes through the vertex of the cones.

In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and theline at infinity), a single point (in fact, twocomplex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines).

All these degenerate conics may occur inpencils of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equationsf = 0 andg = 0, the conics of equationsaf +bg = 0 form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choosef andg so that the given degenerate conic belongs to the pencil they determine.

The conic section with equation${\displaystyle x^2-y^2=0}$ is degenerate as its equation can be written as${\displaystyle (x-y)(x+y)=0}$, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case${\displaystyle a=1,b=0}$ in thepencil ofhyperbolas of equations${\displaystyle a(x^2-y^2)-b=0.}$ The limiting case${\displaystyle a=0,b=1}$ is an example of a degenerate conic consisting of twice the line at infinity.

Similarly, the conic section with equation${\displaystyle x^2+y^2=0}$, which has only one real point, is degenerate, as${\displaystyle x^2+y^2}$ is factorable as${\displaystyle (x+iy)(x-iy)}$ over thecomplex numbers. The conic consists thus of twocomplex conjugate lines that intersect in the unique real point,${\displaystyle (0,0)}$, of the conic.

The pencil of ellipses of equations${\displaystyle a{x}^2+b(y^2-1)=0}$ degenerates, for${\displaystyle a=0,b=1}$, into two parallel lines and, for${\displaystyle a=1,b=0}$, into a double line.

The pencil of circles of equations${\displaystyle a(x^2+y^2-1)-bx=0}$ degenerates for${\displaystyle a=0}$ into two lines, the line at infinity and the line of equation${\displaystyle x=0}$.

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. Any degenerate conic may be transformed by aprojective transformation into any other degenerate conic of the same type.

Over the real affine plane the situation is more complicated. A degenerate real conic may be:

• Two intersecting lines, such as${\displaystyle x^2-y^2=0\iff (x+y)(x-y)=0}$
• Two parallel lines, such as${\displaystyle x^2-1=0\iff (x+1)(x-1)=0}$
• A double line (multiplicity 2), such as${\displaystyle x^2=0}$
• Two intersectingcomplex conjugate lines (only one real point), such as${\displaystyle x^2+y^2=0\iff (x+iy)(x-iy)=0}$
• Two parallel complex conjugate lines (no real point), such as${\displaystyle x^2+1=0\iff (x+i)(x-i)=0}$
• A single line and the line at infinity
• Twice the line at infinity (no real point in theaffine plane)

For any two degenerate conics of the same class, there areaffine transformations mapping the first conic to the second one.

Non-degenerate real conics can be classified as ellipses, parabolas, or hyperbolas by thediscriminant of the non-homogeneous form${\displaystyle A{x}^2+2Bxy+C{y}^2+2Dx+2Ey+F}$, which is the determinant of the matrix

the matrix of the quadratic form in${\displaystyle (x,y)}$. This determinant is positive, zero, or negative as the conic is, respectively, an ellipse, a parabola, or a hyperbola.

Analogously, a conic can be classified as non-degenerate or degenerate according to the discriminant of thehomogeneous quadratic form in${\displaystyle (x,y,z)}$.^{[1][2]: p.16} Here the affine form is homogenized to

${\displaystyle A{x}^2+2Bxy+C{y}^2+2Dxz+2Eyz+F{z}^2;}$

the discriminant of this form is the determinant of the matrix

The conic is degenerate if and only if the determinant of this matrix equals zero. In this case, we have the following possibilities:

• Two intersecting lines (a hyperbola degenerated to its two asymptotes) if and only if (see first diagram).
• Two parallel straight lines (a degenerate parabola) if and only if${\displaystyle detM=0}$. These lines are distinct and real if${\displaystyle D^2+E^2>(A+C)F}$ (see second diagram), coincident if${\displaystyle D^2+E^2=(A+C)F}$, and non-existent in the real plane if.
• A single point (a degenerate ellipse) if and only if${\displaystyle detM>0}$.
• A single line (and the line at infinity) if and only if${\displaystyle A=B=C=0,}$ and${\displaystyle D}$ and${\displaystyle E}$ are not both zero. This case always occurs as a degenerate conic in a pencil ofcircles. However, in other contexts it is not considered as a degenerate conic, as its equation is not of degree 2.

The case of coincident lines occurs if and only if the rank of the 3×3 matrix${\displaystyle Q}$ is 1; in all other degenerate cases its rank is 2.^[3]: p.108

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

Degenerate conics, as with degeneratealgebraic varieties generally, arise as limits of non-degenerate conics, and are important incompactification ofmoduli spaces of curves.

For example, thepencil of curves (1-dimensionallinear system of conics) defined by${\displaystyle x^2+a{y}^2=1}$ is non-degenerate for${\displaystyle a\ne 0}$ but is degenerate for${\displaystyle a=0;}$ concretely, it is an ellipse for${\displaystyle a>0,}$ two parallel lines for${\displaystyle a=0,}$ and a hyperbola with – throughout, one axis has length 2 and the other has length$1/\sqrt{|a|},$ which is infinity for${\displaystyle a=0.}$

Such families arise naturally – given four points ingeneral 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${\displaystyle (\genfrac{}{}{0ex}{}{4}{2,2})=3}$ ways of choosing 2 pairs of points from 4 points (counting via themultinomial coefficient).

External videos
Type I linear system, (Coffman).

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

• ${\displaystyle a>1:}$ hyperbolae opening left and right;
• ${\displaystyle a=1:}$ the parallel vertical lines${\displaystyle x=-1,x=1;}$
• ellipses with a vertical major axis;
• ${\displaystyle a=0:}$ a circle (with radius${\displaystyle \sqrt{2}}$);
• ellipses with a horizontal major axis;
• ${\displaystyle a=-1:}$ the parallel horizontal lines${\displaystyle y=-1,y=1;}$
• hyperbolae opening up and down,
• ${\displaystyle a=\mathrm{\infty }:}$ the diagonal lines${\displaystyle y=x,y=-x;}$

(dividing by${\displaystyle a}$ and taking the limit as${\displaystyle a\to \mathrm{\infty }}$ yields${\displaystyle x^2-y^2=0}$)

• This then loops around to${\displaystyle a>1,}$ since pencils are aprojective line.

Note that this parametrization has a symmetry, where inverting the sign ofa reversesx andy. 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 ageometric 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 theresolvent cubic.

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

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:

• Hyperbolas can degenerate to two intersecting lines (the asymptotes), as in${\displaystyle x^2-y^2=a^2,}$ or to two parallel lines:${\displaystyle x^2-a^2{y}^2=1,}$ or to the double line${\displaystyle x^2-a^2{y}^2=a^2,}$ asa goes to 0.
• Parabolas can degenerate to two parallel lines:${\displaystyle x^2-ay-1=0}$ or the double line${\displaystyle x^2-ay=0,}$ asa goes to 0; but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
• Ellipses can degenerate to two parallel lines:${\displaystyle x^2+a^2{y}^2-1=0}$ or the double line${\displaystyle x^2+a^2{y}^2-a^2=0,}$ asa goes to 0; 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${\displaystyle x^2-a{y}^2-1=0,}$ or to a double line by rotating into each other about a point, as in${\displaystyle x^2-a{y}^2=0,}$ in each case asa goes to 0.
• Two parallel lines can degenerate to a double line by moving into each other, as in${\displaystyle x^2-a^2=0}$ asa goes to 0, but cannot degenerate to non-parallel lines.
• A double line cannot degenerate to the other types.
• Another type of degeneration occurs for an ellipse when the sum of the distances to the foci is mandated to equal the interfocal distance; thus it has semi-minor axis equal to zero and has eccentricity equal to one. The result is aline segment (degenerate because the ellipse is not differentiable at the endpoints) with itsfoci at the endpoints. As anorbit, this is aradial elliptic trajectory.

A general conic isdefined by five points: given five points ingeneral 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 atrapezoid (one pair is parallel) or aparallelogram (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 theparallel postulate.

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

• Conic sections

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