Conic section

The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry.

The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone.

In addition to the eccentricity (e), foci, and directrix, various geometric features and lengths are associated with a conic section.

This form is a specialization of the homogeneous form used in the more general setting of projective geometry (see below).

(or some variation of this) so that the matrix of the conic section has the simpler form,

If the determinant of the matrix of the conic section is zero, the conic section is degenerate.

No continuous arc of a conic can be constructed with straightedge and compass. However, there are several straightedge-and-compass constructions for any number of individual points on an arc.

if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic.

What should be considered as a degenerate case of a conic depends on the definition being used and the geometric setting for the conic section. There are some authors who define a conic as a two-dimensional nondegenerate quadric. With this terminology there are no degenerate conics (only degenerate quadrics), but we shall use the more traditional terminology and avoid that definition.

When viewed from the perspective of the complex projective plane, the degenerate cases of a real quadric (i.e., the quadratic equation has real coefficients) can all be considered as a pair of lines, possibly coinciding. The empty set may be the line at infinity considered as a double line, a (real) point is the intersection of two complex conjugate lines and the other cases as previously mentioned.

A generalization of a non-degenerate conic in a projective plane is an oval. An oval is a point set that has the following properties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of the oval there exists a unique tangent line.

Generalizing the focus properties of conics to the case where there are more than two foci produces sets called generalized conics.

The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. The classification mostly arises due to the presence of a quadratic form (in two variables this corresponds to the associated discriminant), but can also correspond to eccentricity.