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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, despite the fact that the concept is widely used and defined (if needed) in each specific situation.
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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. In most scenarios, a given class of objects may be defined by several different
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vertices and zero area. 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. If all three vertices are equal, all three angles are
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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.
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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
517:{\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\}}
87:, which makes its dimension one. This is similar to the case of a circle, whose dimension shrinks from two to zero as it degenerates into a point. As another example, the
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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
83:
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
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sides looks like a polygon with fewer sides. In the case of triangles, this definition coincides with the one that has been given above.
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The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the
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of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "
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788:: valid in a general abstract mathematical sense, but not part of the original Euclidean conception of polygons.
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can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common
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where the axis of revolution passes through the center of the generating circle, rather than outside it.
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th degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate
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When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a
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is degenerate if and only if it has singular points (e.g., point, line, intersecting lines).
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if at least one side length or angle is zero. Equivalently, it becomes a "line segment".
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This article is about degeneracy in mathematics. For the degeneracy of a graph, see
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are non-generic and non-degenerate. In fact, degenerate cases often correspond to
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Often, the degenerate cases are the exceptional cases where changes to the usual
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115:. However, not all non-generic or special cases are degenerate. For example,
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The three types of degenerate triangles, all of which contain zero area.
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In contexts where self-intersection is allowed, a double-covered
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A torus degenerates to a circle when its minor radius goes to 0.
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is a "flat" triangle in the sense that it is contained in a
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A circle can be thought of as a degenerate ellipse, as the
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A degenerate case thus has special features which makes it
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Limiting case which is different from the rest of the class
861:. Usually any such degeneracy indicates some underlying
358:, there is a bounded, axis-aligned degenerate rectangle
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An ellipse can also degenerate into a single point.
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769:A set containing a single point is a degenerate
690:is degenerate if either two adjacent facets are
698:, this is equivalent to saying that all of its
57:" is the condition of being a degenerate case.
8:
594:{\displaystyle \mathbf {x} \triangleq \left}
351:{\displaystyle S\subseteq \{1,2,\ldots ,n\}}
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694:or two edges are aligned. In the case of a
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301:A line segment is a degenerate case of a
223:can be viewed as a degenerate case of an
646:is the number of elements of the subset
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1002:"Mathwords: Degenerate Conic Sections"
216:lines also form a degenerate parabola.
784:can be viewed as degenerate cases of
642:). The number of degenerate sides of
170:of degree two) that fails to be an
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795:which can only take one value has a
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246:approaches 0 and the foci merge.
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235:go to the endpoints, and the
197:if the parabola resides on a
36:Degeneracy (disambiguation)
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949:"Definition of DEGENERACY"
891:Pathological (mathematics)
853:in the eigenvalues of the
838:is a multiple root of the
193:is a degenerate case of a
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64:and the side lengths of a
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876:Degeneracy (graph theory)
840:characteristic polynomial
308:For any non-empty subset
32:Degeneracy (graph theory)
859:degenerate energy levels
822:, since generically the
814:is sometimes said to be
158:A degenerate conic is a
18:Degenerate (mathematics)
978:"Mathwords: Degenerate"
953:www.merriam-webster.com
797:degenerate distribution
209:, with infinite radius.
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34:. For other uses, see
1031:Mathematical concepts
929:mathworld.wolfram.com
886:Trivial (mathematics)
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125:equilateral triangles
855:Hamiltonian operator
801:Dirac delta function
776:Objects such as the
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923:Weisstein, Eric W.
759:for other examples.
622:are constant (with
168:polynomial equation
133:configuration space
121:isosceles triangles
93:system of equations
70:degenerate triangle
591:
514:
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231:goes to zero, the
203:inversive geometry
1006:www.mathwords.com
982:www.mathwords.com
847:quantum mechanics
688:convex polyhedron
681:Convex polyhedron
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172:irreducible curve
162:(a second-degree
135:. For example, a
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278:A degenerate
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227:in which the
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148:Conic section
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129:singularities
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51:limiting case
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1009:. Retrieved
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985:. Retrieved
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956:. Retrieved
952:
943:
932:. Retrieved
928:
925:"Degenerate"
851:multiplicity
830:
828:roots of an
824:
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673:
631:
624:
284:line segment
244:eccentricity
239:goes to one.
237:eccentricity
221:line segment
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113:special case
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102:
89:solution set
74:
69:
59:
54:
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40:
849:, any such
818:if it is a
696:tetrahedron
164:plane curve
143:In geometry
109:non-generic
81:cardinality
43:mathematics
1011:2019-11-29
987:2019-11-29
958:2019-11-29
934:2019-11-29
902:References
836:eigenvalue
816:degenerate
812:polynomial
291:undefined.
258:asymptotes
97:parameters
55:degeneracy
771:continuum
764:Elsewhere
571:…
537:≜
501:∉
494:for
474:≤
461:≤
437:∈
430:for
382:∈
369:≜
337:…
319:⊆
303:rectangle
296:Rectangle
288:collinear
254:hyperbola
77:dimension
1025:Category
870:See also
863:symmetry
786:polygons
710:of zero.
700:vertices
692:coplanar
638:for all
280:triangle
265:Triangle
214:parallel
195:parabola
66:triangle
782:monogon
225:ellipse
111:, or a
79:or the
736:Sphere
722:sphere
708:volume
524:where
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207:circle
184:circle
62:angles
810:of a
778:digon
750:Other
743:point
704:plane
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180:point
91:of a
49:is a
808:root
780:and
755:See
601:and
233:foci
212:Two
191:line
189:The
123:and
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