Екінші дәрежелі қисықтардың тоқырауы

2nd degree plane curve which is reducible

In geometry, a degenerate conic is a conic (a second degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more generally over an algebraically closed field) 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 (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate conics may occur in pencils of conics. That is, if two real non degenerated conics are defined by quadratic polynomial equations 1=f = 0 and 1=g = 0, the conics of equations 1=af + bg = 0 form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose f and g so that the given degenerate conic belongs to the pencil they determine.

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 .

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.

A general conic is defined by five points: given five points in general position, 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.