mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

s and the side lengths of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

or the

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

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, which makes its dimension one. This is similar to the case of a circle, whose dimension shrinks from two to zero as it degenerates into a point. As another example, the

solution set In mathematics, the solution set of a system of equations or inequality is the set of all its solutions, that is the values that satisfy all equations and inequalities. Also, the solution set or the truth set of a statement or a predicate is t ...

of a

system of equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...

that depends on

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the

is said to be degenerate. For some classes of composite objects, the degenerate cases depend on the properties that are specifically studied. In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by several different systems of equations, and these different systems of equations may lead to different degenerate cases, while characterizing the same non-degenerate cases. This may be the reason for which there is no general definition of degeneracy, despite the fact that the concept is widely used and defined (if needed) in each specific situation. A degenerate case thus has special features which makes it non-generic, or a

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of .Brown, James Robert.� ...

. However, not all non-generic or special cases are degenerate. For example,

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

s and

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

is degenerate if and only if it has singular points (e.g., point, line, intersecting lines).

In geometry

Conic section

A degenerate conic is a

(a second-degree

plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...

, defined by a

polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For example, x^5-3x+1=0 is a ...

of degree two) that fails to be an irreducible curve. * A point is a degenerate

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, namely one with radius 0. * The line is a degenerate case of a

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

if the parabola resides on a

tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

. In

inversive geometry In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...

, a line is a degenerate case of a

, with infinite radius. * Two parallel lines also form a degenerate parabola. * A

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

can be viewed as a degenerate case of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

in which the

semiminor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...

goes to zero, the

foci Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film * ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel * ''Focus'' (2015 film), a 201 ...

go to the endpoints, and the

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

goes to one. *A circle can be thought of as a degenerate ellipse, as the

approaches 0 and the foci merge. * An ellipse can also degenerate into a single point. * A

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

Triangle

A degenerate

is a "flat" triangle in the sense that it is contained in a

. It has thus

collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

vertices and zero area. If the three vertices are all 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.

Rectangle

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

with one pair of opposite sides of length zero degenerates to a line segment, with zero area. If both of the rectangle's pairs of opposite sides have length zero, the rectangle degenerates to a point.

Hyperrectangle

hyperrectangle In geometry, a hyperrectangle (also called a box, hyperbox, k-cell or orthotopeCoxeter, 1973), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient cond ...

is the -dimensional analog of a rectangle. If its sides along any of the axes has length zero, it degenerates to a lower-dimensional hyperrectangle, all the way down to a point if the sides aligned with every axis have length zero.

Convex polygon

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

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.

Convex polyhedron

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

is degenerate if either two adjacent facets are

coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...

or two edges are aligned. In the case of a

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, this is equivalent to saying that all of its vertices lie in the same plane, giving it a

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of zero.

Standard torus

* In contexts where self-intersection is allowed, a double-covered

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

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.

Sphere

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

Other

See

general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that a ...

for other examples.

Elsewhere

* A set containing a single point is a degenerate continuum. * Objects such as the

digon In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...

and

monogon In geometry, a monogon, also known as a henagon, is a polygon with one Edge (geometry), edge and one Vertex (geometry), vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386 ...

can be viewed as degenerate cases of

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s: valid in a general abstract mathematical sense, but not part of the original Euclidean conception of polygons. * A

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

which can only take one value has a

degenerate distribution In probability theory, a degenerate distribution on a measure space (E, \mathcal, \mu) is a probability distribution whose support is a null set with respect to \mu. For instance, in the -dimensional space endowed with the Lebesgue measure, an ...

; if that value is the real number 0, then its probability density is the

Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

. * A

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

of a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

is sometimes said to be ''degenerate'' if it is a

multiple root In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...

, since generically the roots of an th degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

is a multiple root of the

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

. * In

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, any such multiplicity in the eigenvalues of the

Hamiltonian operator In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalu ...

gives rise to

degenerate energy level In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give th ...

s. Usually any such degeneracy indicates some underlying

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

in the system.

References

{{DEFAULTSORT:Degeneracy (Mathematics) Mathematical concepts