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Concept
Degenerate conic
Résumé
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 f = 0 and g = 0, the conics of equations 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 .
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