In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class.

• A point is a degenerate circle, namely one with radius 0. The circle is a degenerate form of an ellipse, namely one with eccentricity 0.
• The line is a degenerate form of a parabola if the parabola resides on a tangent plane.
• A segment is a degenerate form of a rectangle, if this has a side of length 0.
• A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.
• A set containing a single point is a degenerate continuum.
• A random variable which can only take one value is degenerate.
• See "general position" for other examples.

Another usage of the word comes in eigenproblems: a degenerate eigenvalue is one that has more than one linearly independent eigenvector.

Degenerate rectangle

For any non-empty subset $S$ of the indices $\{1,\; 2,\; ...,\; n\},$ a bounded, axis-aligned degenerate rectangle $R$ is a subset of $mathcal\{R\}^n$ of the following form:

$R\; =\; left\{mathbf\{x\}\; :\; x\_i\; =\; c\_i\; (mathrm\{for\}\; iin\; S)\; mathrm\{and\}\; a\_i\; leq\; x\_i\; leq\; b\_i\; (mathrm\{for\}\; i\; notin\; S)right\}$

where $mathbf\{x\}=\; [x\_1,\; x\_2,\; ldots,\; x\_n]$ 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).

See also

• Trivial (mathematics)