# Degeneracy (mathematics)

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.

The definitions of many classes of composite or structured objects include (often implicitly) 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.

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, and its dimension is thus one. Similarly, the solution set of a system of equations that depends on parameters 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 solution set 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. Commonly, 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, although 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. However, not all non-generic cases are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. Frequently, degenerate cases correspond to singularities either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points.

## In geometry

### Conic section

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.

### Triangle

• A degenerate triangle has collinear vertices and zero area, and thus coincides with a segment covered twice (if the three vertices are not all equal; otherwise, the triangle degenerates to a single point). 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.

### Rectangle

• A line segment is a degenerate case of a rectangle which has a side of length 0.
• For any non-empty subset $S\subseteq \{1,2,\ldots ,n\}$ , there is a bounded, axis-aligned degenerate rectangle
$R\triangleq \left\{\mathbf {x} \in \mathbb {R} ^{n}:x_{i}=c_{i}\ ({\text{for }}i\in S){\text{ and }}a_{i}\leq x_{i}\leq b_{i}\ ({\text{for }}i\notin S)\right\}$ where $\mathbf {x} \triangleq [x_{1},x_{2},\ldots ,x_{n}]$ and $a_{i},b_{i},c_{i}$ are constant (with $a_{i}\leq b_{i}$ 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).

### Convex polygon

• 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.

### Standard torus

• In contexts where self-intersection is allowed, 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.