# 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.

### 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.
${\displaystyle R\triangleq \left\{{\mathbf {x} }\in {\mathbb {R} }^{n}:x_{i}=c_{i}\ ({\text{for }}i\in S){\text{ and }}a_{i}\leq x_{i}\leq b_{i}\ ({\text{for }}i\notin S)\right\}}$

where ${\displaystyle {\mathbf {x} }\triangleq [x_{1},x_{2},\ldots ,x_{n}]}$ and ${\displaystyle a_{i},b_{i},c_{i}}$ are constant (with ${\displaystyle a_{i}\leq b_{i}}$ for all ${\displaystyle i}$). The number of degenerate sides of ${\displaystyle R}$ is the number of elements of the subset ${\displaystyle S}$. Thus, there may be as few as one degenerate "side" or as many as ${\displaystyle n}$ (in which case ${\displaystyle R}$ reduces to a singleton point).

### 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.

## Elsewhere

• A set containing a single point is a degenerate continuum.
• Similarly, roots of a polynomial are said to be degenerate if they coincide, since generically the n roots of an nth 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.