Degenerate conic
In geometry, a degenerate conic is a conic that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers as the product of two linear polynomials.
Using the alternative definition of the conic as the intersection in three-dimensional space of a plane and a double cone, 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, a single point, or the null set.
All these degenerate conics may occur in pencils 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 is degenerate as its equation can be written as, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case in the pencil of hyperbolas of equations The limiting case is an example of a degenerate conic consisting of twice the line at infinity.Similarly, the conic section with equation, which has only one real point, is degenerate, as is factorable as over the complex numbers. The conic consists thus of two complex conjugate lines that intersect in the unique real point,, of the conic.
The pencil of ellipses of equations degenerates, for, into two parallel lines and, for, into a double line.
The pencil of circles of equations degenerates for into two lines, the line at infinity and the line of equation.
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 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
- Two parallel lines, such as
- A double line, such as
- Two intersecting complex conjugate lines, such as
- Two parallel complex conjugate lines, such as
- A single line and the line at infinity
- Twice the line at infinity
Discriminant
Non-degenerate real conics can be classified as ellipses, parabolas, or hyperbolas by the discriminant of the non-homogeneous form, which is the determinant of the matrixthe matrix of the quadratic form in. 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. Here the affine form is homogenized to
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 if and only if .
- Two parallel straight lines if and only if. These lines are distinct and real if , coincident if, and non-existent in the real plane if.
- A single point if and only if.
- A single line if and only if and and are not both zero. This case always occurs as a degenerate conic in a pencil of circles. However, in other contexts it is not considered as a degenerate conic, as its equation is not of degree 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. 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
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 defined by is non-degenerate for but is degenerate for concretely, it is an ellipse for two parallel lines for and a hyperbola with – throughout, one axis has length 2 and the other has length which is infinity for
Such families arise naturally – given four points in general linear position, there is a pencil of conics through them, of which three are degenerate, each consisting of a pair of lines, corresponding to the ways of choosing 2 pairs of points from 4 points.
For example, given the four points the pencil of conics through them can be parameterized as yielding the following pencil; in all cases the center is at the origin:
- hyperbolae opening left and right;
- the parallel vertical lines
- ellipses with a vertical major axis;
- a circle ;
- ellipses with a horizontal major axis;
- the parallel horizontal lines
- hyperbolae opening up and down,
- the diagonal lines
- This then loops around to since pencils are a projective line.
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.
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:
- Hyperbolas can degenerate to two intersecting lines, as in or to two parallel lines: or to the double line as a goes to 0.
- Parabolas can degenerate to two parallel lines: or the double line 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: or the double line 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.
- Two intersecting lines can degenerate to two parallel lines, by rotating until parallel, as in or to a double line by rotating into each other about a point, as in in each case as a goes to 0.
- Two parallel lines can degenerate to a double line by moving into each other, as in 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 with its foci at the endpoints. As an orbit, this is a radial elliptic trajectory.
Points to define
Given four points in general linear position, there are exactly three pairs of lines passing through them, which will in general be intersecting, unless the points form a trapezoid or a parallelogram.
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.