Matrix representation of conic sections

In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.

Conic sections (including degenerate ones) are the sets of points whose coordinates satisfy a second-degree polynomial equation in two variables,

This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as^[1]

The sum of the first three terms of this equation, namely

The quadratic equation can also be written as

where $\mathbf {x}$ is the homogeneous coordinate vector in three variables restricted so that the last variable is 1, i.e.,

and where $A_{Q}$ is the matrix

The 2 × 2 upper left submatrix (a matrix of order 2) of $A Q$ , obtained by removing the third (last) row and third (last) column from $A Q$ is the matrix of the quadratic form. The above notation $A 33$ is used in this article to emphasize this relationship.

$Q$ is a if and only if $\det A_{33}<0$ ,

hyperbola

$Q$ is a if and only if $\det A_{33}=0$ , and

parabola

$Q$ is an if and only if $\det A_{33}>0$ .

ellipse

Proper (non-degenerate) and degenerate conic sections can be distinguished^[5]^[6] based on the determinant of $A Q$ :

If $\det A_{Q}=0$ , the conic is degenerate.

If $\det A_{Q}\neq 0$ so that $Q$ is not degenerate, we can see what type of conic section it is by computing the minor, $\det A_{33}$ :

In the case of an ellipse, we can distinguish the special case of a circle by comparing the last two diagonal elements corresponding to the coefficients of $x 2$ and $y 2$ :

Moreover, in the case of a non-degenerate ellipse (with $\det A_{33}>0$ and $\det A_{Q}\neq 0$ ), we have a real ellipse if $(A+C)\det A_{Q}<0$ but an imaginary ellipse if $(A+C)\det A_{Q}>0$ . An example of the latter is $x^{2}+y^{2}+10=0$ , which has no real-valued solutions.

If the conic section is degenerate ( $\det A_{Q}=0$ ), $\det A_{33}$ still allows us to distinguish its form:

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

If $λ 1$ and $λ 2$ have the same algebraic sign, then $Q$ is a real ellipse, imaginary ellipse or real point if $K$ has the same sign, has the opposite sign or is zero, respectively.

If $λ 1$ and $λ 2$ have opposite algebraic signs, then $Q$ is a hyperbola or two intersecting lines depending on whether $K$ is nonzero or zero, respectively.

Conic section § General Cartesian form

Quadratic form (statistics)

Ayoub, A. B. (1993), "The central conic sections revisited", , 66 (5): 322–325, doi:10.1080/0025570x.1993.11996157

Mathematics Magazine

Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999), Geometry, Cambridge University Press, 978-0-521-59787-6

ISBN

Lawrence, J. Dennis (1972), A Catalog of Special Plane Curves, Dover

Ostermann, Alexander; Wanner, Gerhard (2012), Geometry by its History, Springer, :10.1007/978-3-642-29163-0, ISBN 978-3-642-29163-0

doi

Pettofrezzo, Anthony (1978) [1966], , Dover, ISBN 978-0-486-63634-4

Matrices and Transformations

Spain, Barry (2007) [1957], Analytical Conics, Dover, 978-0-486-45773-4