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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a degenerate conic is a
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
(a second-degree
plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
, defined by a
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (or more generally over an algebraically closed field) as the product of two linear polynomials.Some authors consider conics without real points as degenerate, but this is not a commonly accepted convention. Using the alternative definition of the conic as the intersection in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
of a plane and a double
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
, 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 the
line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The ...
), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate conics may occur in
pencils A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...
of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equations and , the conics of equations form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose and so that the given degenerate conic belongs to the pencil they determine.


Examples

The conic section with equation x^2-y^2 = 0 is degenerate as its equation can be written as (x-y)(x+y)= 0, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case a=1, b=0 in the
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...
of
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s of equations a(x^2-y^2) - b=0. The limiting case a=0, b=1 is an example of a degenerate conic consisting of twice the line at infinity. Similarly, the conic section with equation x^2 + y^2 = 0, which has only one real point, is degenerate, as x^2+y^2 is factorable as (x+iy)(x-iy) over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. The conic consists thus of two complex conjugate lines that intersect in the unique real point, (0,0), of the conic. The pencil of ellipses of equations ax^2+b(y^2-1)=0 degenerates, for a=0, b=1, into two parallel lines and, for a=1, b=0, into a double line. The pencil of circles of equations a(x^2+y^2-1) - bx =0 degenerates for a=0 into two lines, the line at infinity and the line of equation x=0.


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. Any degenerate conic may be transformed by a
projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
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 x^2-y^2 = 0 \Leftrightarrow (x+y)(x-y) = 0 * Two parallel lines, such as x^2-1 = 0 \Leftrightarrow (x+1)(x-1) = 0 * A double line (multiplicity 2), such as x^2 = 0 * Two intersecting complex conjugate lines (only one real point), such as x^2+y^2 = 0 \Leftrightarrow (x+iy)(x-iy) = 0 * Two parallel complex conjugate lines (no real point), such as x^2+1 = 0 \Leftrightarrow (x+i)(x-i) = 0 * A single line and the line at infinity * Twice the line at infinity (no real point in the
affine plane In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are *Euclidean planes, which are affine planes over the real number, reals equipped with a metric (mathematics), metric, the Euclidean dista ...
) For any two degenerate conics of the same class, there are affine transformations mapping the first conic to the second one.


Discriminant

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 :M=\begin A & B \\ B & C \\ \end, the matrix of the quadratic form in (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 the ''homogeneous'' quadratic form in (x,y,z). 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 :Q=\begin A & B & D \\ B & C & E \\ D & E & F \\ \end. 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 \det M< 0 (see first diagram). * Two parallel straight lines (a degenerate parabola) if and only if \det M= 0. These lines are distinct and real if D^2+E^2>(A+C)F (see second diagram), coincident if D^2+E^2=(A+C)F, and non-existent in the real plane if D^2+E^2<(A+C)F. * A single point (a degenerate ellipse) if and only if \det M> 0. * A single line (and the line at infinity) if and only if A=B=C=0, and D and E are not both zero. This case always occurs as a degenerate conic in a pencil of
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s. 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 Q is 1; in all other degenerate cases its rank is 2.


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 with a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
. Degeneracy occurs when the plane contains the
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
generally, arise as limits of non-degenerate conics, and are important in compactification of moduli spaces of curves. For example, the
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...
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, 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 In Euclidean and projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). There are additional subtleties for conics that do not exist for lines, and thu ...
, four points leave one parameter free), of which three are degenerate, each consisting of a pair of lines, corresponding to the \textstyle ways of choosing 2 pairs of points from 4 points (counting via the
multinomial coefficient In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer ...
). 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:A simpler parametrization is given by ax^2+(1-a)y^2=1, which are the
affine combination In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ ...
s 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.
* 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); * -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 , 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 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 In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: :P(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0. In each case: * The coefficients of the resolvent cubic can be obta ...
.
Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
is the special case of
Pascal's theorem In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined ...
, 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: * Hyperbolas 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 the double line x^2-a^2y^2=a^2, as ''a'' goes to 0. * Parabolas can degenerate to two parallel lines: x^2-ay-1=0 or the double line x^2-ay=0, as ''a'' 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: x^2+a^2y^2-1=0 or the double line x^2+a^2y^2-a^2=0, as ''a'' 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 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, in each case as ''a'' goes to 0. * Two parallel lines can degenerate to a double line by moving into each other, as in x^2-a^2=0 as ''a'' 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 a line segment (degenerate because the ellipse is not differentiable at the endpoints) with its foci at the endpoints. As an
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
, this is a
radial elliptic trajectory In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it ...
.


Points to define

A general conic is defined by five points: given five points in
general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
, 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 A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
(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 In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segmen ...
. Given two distinct points, there is a unique double line through them.


Notes


References

* * * * * * * * {{refend Conic sections