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,

are both invariant with respect to rotation of axes and translation of the plane (movement of the origin).

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

is not degenerate, we can see what type of conic section it is by computing the minor,

still allows us to distinguish its form: The case of coincident lines occurs if and only if the rank of the 3 × 3 matrix

This property can be used to calculate the coordinates of the center, which can be shown to be the point where the gradient of the quadratic function Q vanishes—that is,[8]

An alternative approach that uses the matrix form of the quadratic equation is based on the fact that when the center is the origin of the coordinate system, there are no linear terms in the equation.

The condition for (x0, y0) to be the conic's center (xc, yc) is that the coefficients of the linear x* and y* terms, when this equation is multiplied out, are zero.

This calculation can also be accomplished by taking the first two rows of the associated matrix AQ, multiplying each by (x, y, 1)⊤ and setting both inner products equal to 0, obtaining the following system:

The standard form of the equation of a central conic section is obtained when the conic section is translated and rotated so that its center lies at the center of the coordinate system and its axes coincide with the coordinate axes.

The rotation by angle α can be carried out by diagonalizing the matrix A33.

are the eigenvalues of the matrix A33, the centered equation can be rewritten in new variables x' and y' as[9]

From here we get a and b, the lengths of the semi-major and semi-minor axes in conventional notation.

[10] By the principal axis theorem, the two eigenvectors of the matrix of the quadratic form of a central conic section (ellipse or hyperbola) are perpendicular (orthogonal to each other) and each is parallel to (in the same direction as) either the major or minor axis of the conic.

The eigenvector having the smallest eigenvalue (in absolute value) corresponds to the major axis.

[11] Specifically, if a central conic section has center (xc, yc) and an eigenvector of A33 is given by v(v1, v2) then the principal axis (major or minor) corresponding to that eigenvector has equation,

Two or no vertices are obtained for each axis, since, in the case of the hyperbola, the minor axis does not intersect the hyperbola at a point with real coordinates.

However, from the broader view of the complex plane, the minor axis of an hyperbola does intersect the hyperbola, but at points with complex coordinates.

If the conic is non-degenerate, the conjugates of a point always form a line and the polarity defined by the conic is a bijection between the points and lines of the extended plane containing the conic (that is, the plane together with the points and line at infinity).

Just as p uniquely determines its polar line (with respect to a given conic), so each line determines a unique pole p. Furthermore, a point p is on a line L which is the polar of a point r, if and only if the polar of p passes through the point r (La Hire's theorem).

[15] Thus, this relationship is an expression of geometric duality between points and lines in the plane.

Several familiar concepts concerning conic sections are directly related to this polarity.

The center of a non-degenerate conic can be identified as the pole of the line at infinity.

Also, the polar line of a focus of the conic is its corresponding directrix.

Finally, if L does not intersect Q then p has no tangent lines passing through it and it is called an interior or inner point.

[17] The equation of the tangent line (in homogeneous coordinates) at a point p on the non-degenerate conic Q is given by,

If p is an exterior point, first find the equation of its polar (the above equation) and then the intersections of that line with the conic, say at points s and t. The polars of s and t will be the tangents through p. Using the theory of poles and polars, the problem of finding the four mutual tangents of two conics reduces to finding the intersection of two conics.