# Degeneracy (mathematics)

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In mathematics, a **degenerate case** is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. **Degeneracy** is the condition of being a degenerate case.

A degenerate case thus has special features, which depart from the properties that are generic in the wider class, and which would be lost under an appropriate small perturbation.

## In geometry

### Conic section

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A degenerate conic is a conic section (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) 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.
- 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.
- 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.

### Triangle

- A degenerate triangle has collinear vertices and zero area, and thus coincides with a segment covered twice.

### Rectangle

- A segment is a degenerate case of a rectangle, if this has a side of length 0.

where and are constant (with for all ). The number of degenerate sides of is the number of elements of the subset . Thus, there may be as few as one degenerate "side" or as many as (in which case reduces to a singleton point).

### Polyhedron

- A polyhedron such as a tetrahedron or other pyramid is degenerate if all of its vertices lie in the same plane, giving it zero volume.

### Standard torus

- A sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it.

### Sphere

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

### Other

- See general position for other examples.

## Elsewhere

- A set containing a single point is a degenerate continuum.

- A random variable which can only take one value has a degenerate distribution; its probability density is the Dirac Delta function at one point.

- Similarly, roots of a polynomial are said to be
*degenerate*if they coincide, since generically the*n*roots of an*n*th degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiple coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.

- 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.

## See also

- Degeneracy (graph theory)
- Degenerate form
- Trivial (mathematics)
- Pathological (mathematics)
- Vacuous truth