Degeneracy (mathematics)
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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. Also it 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.
- See "general position" for other examples.
Degenerate rectangle
For any non-empty subset of the indices [\,] a bounded degenerate rectangle [R] is a subset of [\mathcal^n] of the following form:
[R = \left\ : x_i = c_i \ (\mathrm \ i\in S) \ \mathrm \ a_i \leq x_i \leq b \ (\mathrm \ i \notin S)\right\}]
where [\mathbf= [x_1, x_2, ldots, x_n]]. 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: degeneracy, Trivial (mathematics).
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