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  • 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 Asymptote s.

  • A set containing a single point is a degenerate Continuum .

  • 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 of the indices \{1, 2, ..., n\}, a bounded degenerate rectangle R is a subset of \mathcal{R}^n of the following form:

R = \left\{\mathbf{x} : x_i = c_i \ (\mathrm{for} \ i\in S) \ \mathrm{and} \ a_i \leq x_i \leq b \ (\mathrm{for} \ i
otin S) ight\}

where \mathbf{x}= 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) .