Математикада дегенеративті жағдайлар

Limiting case which is different from the rest of the class
degeneracy in mathematics

In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a degenerate triangle if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line,

A degenerate conic is a conic section (a second degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. A point is a degenerate circle, namely one with radius 0. The line is a degenerate case of a parabola if the parabola resides on a tangent plane. In inversive geometry, a line is a degenerate case of a circle, with infinite radius. Two parallel lines also form a degenerate parabola. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one. A circle can be thought of as a degenerate ellipse, as the eccentricity approaches 0 and the foci merge. An ellipse can also degenerate into a single point. A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common asymptotes.

A degenerate triangle is a "flat" triangle in the sense that it is contained in a line segment. It has thus collinear vertices and zero area. If the three vertices are pairwise distinct, it has two 0° angles and one 180° angle. If two vertices are equal, it has one 0° angle and two undefined angles. If all three vertices are equal, all three angles are undefined.

A line segment is a degenerate case of a rectangle which has a side of length 0. For any non empty subset , there is a bounded, axis aligned degenerate rectangle where and ai, bi, ci are constant (with ai ≤ bi for all i). The number of degenerate sides of R is the number of elements of the subset S. Thus, there may be as few as one degenerate "side" or as many as n (in which case R reduces to a singleton point).

A convex polygon is degenerate if at least two consecutive sides coincide at least partially, or at least one side has zero length, or at least one angle is 180°. Thus a degenerate convex polygon of n sides looks like a polygon with fewer sides. In the case of triangles, this definition coincides with the one that has been given above.

A convex polyhedron is degenerate if either two adjacent facets are coplanar or two edges are aligned. In the case of a tetrahedron, this is equivalent to saying that all of its vertices lie in the same plane, giving it a volume of zero.

In contexts where self intersection is allowed, a double covered sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it. A torus degenerates to a circle when its minor radius goes to 0.

When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a point.

See general position for other examples.

A set containing a single point is a degenerate continuum. Objects such as the digon and monogon can be viewed as degenerate cases of polygons: valid in a general abstract mathematical sense, but not part of the original Euclidean conception of polygons. A random variable which can only take one value has a degenerate distribution; if that value is the real number 0, then its probability density is the Dirac delta function. A root of a polynomial is sometimes said to be degenerate if it is a multiple root, since generically the n roots of an nth degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue is a multiple root of the characteristic polynomial. In quantum mechanics, any such multiplicity in the eigenvalues of the Hamiltonian operator gives rise to degenerate energy levels. Usually any such degeneracy indicates some underlying symmetry in the system.