Degenerate conic
Encyclopedia
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.
is not degenerate (it defines a hyperbola
), but
is degenerate because it is reducible – it factors as 
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
is not degenerate (it defines a circle
) but
is degenerate – it defines a point, 
not a curve, and 
is likewise degenerate, defining the empty set. Note that over the complex numbers 
factors as 
and is degenerate because reducible, while 
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).
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.
s (if it has any at all). This can occur in two ways:
of their imaginary part, considered as a quadratic form in
(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 
is zero, where the affine equation
(factors of 2 for cross terms) is homogenized to
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
of moduli spaces of curves.
For example, the pencil
of curves (1-dimensional linear system of conics) 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 (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
ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient).
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:A simpler parametrization is given by 
which are the affine combinations of the equations 
and 
corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of 
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.
Pappus's hexagon theorem
is the special case of Pascal's theorem
, when a conic degenerates to two lines.
In the real affine plane:
Degenerate conics can degenerate further to more special degenerate conics, as indicated by the dimensions of the spaces and points at infinity.
, 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.
, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a degenerate conic is a conic (degree-2 plane curve
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....
, 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, is not degenerate (it defines a hyperbolaHyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
), but
is degenerate because it is reducible – it factors as 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
is not degenerate (it defines a circleCircle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
) but
is degenerate – it defines a point, not a curve, and is likewise degenerate, defining the empty set. Note that over the complex numbers factors as and is degenerate because reducible, while 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
– a 4-dimensional space - Two parallel lines, such as
– a 3-dimensional space - A single doubled line (multiplicity 2), such as
– 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 pointReal point
A point in the complex projective plane is called real if there exists a complex number z such that za, zb and zc are all real numbers.This definition can be widened to complex projective space and complex projective hyperspaces as follows:...
s (if it has any at all). This can occur in two ways:
- A single double point, such as
- No points, such as
– an imaginary ellipse.
Discriminant
Just as non-degenerate real conics can be classified by the discriminantDiscriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
of their imaginary part, considered as a quadratic form in
(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 is zero, where the affine equation(factors of 2 for cross terms) is homogenized to
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 compactificationCompactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of moduli spaces of curves.
For example, the pencil
Pencil (mathematics)
A pencil in projective geometry is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a projective plane....
of curves (1-dimensional linear system of conics) 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 (no three on a line), there is a pencil of conics through them (five points determine a conic
Five points determine a conic
In geometry, just as two points determine a line , five points determine a conic . There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.Formally, given any five points in the plane in general...
, four points leave one parameter free), 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 (counting via the multinomial coefficient).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:A simpler parametrization is given by which are the affine combinations of the equations and corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of
-
hyperbolae opening left and right; -
the parallel vertical lines 
-
ellipses with a vertical major axis; -
a circle (with radius 
); -
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.
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.
Pappus's hexagon theorem
Pappus's hexagon theorem
In mathematics, Pappus's hexagon theorem 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 aC, Bc and bC are collinear...
is the special case of Pascal's theorem
Pascal's theorem
In projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.- Related results :This 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
or to two parallel lines: 
or to double line: 
- parabolae can degenerate to two parallel lines:
or a double line 
but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines. - ellipses can degenerate to two parallel lines:
or a double line 
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
or to a double line by rotating into each other about a point, as in 
- Two parallel lines can degenerate to a double line by moving into each other, as in
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 positionGeneral position
In algebraic 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 possible...
, 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
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
(one pair is parallel) or a parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
(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
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...
.
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, has its semiminor axis b go to zero and thus has its eccentricity go to one. The result is a line segmentLine segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
(degenerate because the ellipse is not differentiable at the endpoints) with its foci
Focus (geometry)
In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
at the endpoints. As an orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
, this is a radial elliptic trajectory.