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Degeneracy (mathematics)
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<h2 id="in-geometry">In geometry</h2> <h3>Conic section</h3> <p class="note">Main article: <a href="/facts/Degenerate_conic/9Ek4ja1g">Degenerate conic</a></p> <p>A degenerate conic is a <a href="/facts/Conic_section/teqgLptX">conic section</a> (a second-degree <a href="/facts/Plane_curve/huCfWa4K">plane curve</a>, defined by a <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equation</a> of degree two) that fails to be an <a href="/facts/Irreducible_variety/MxWWaCM7">irreducible curve</a>. </p> <ul><li>A <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> is a degenerate <a href="/facts/Circle/9fy5ggKn">circle</a>, namely one with radius 0.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li> <li>The <a href="/facts/Line_(mathematics)/gJqsr4Gj">line</a> is a degenerate case of a <a href="/facts/Parabola/Cweo3wgw">parabola</a> if the parabola resides on a <a href="/facts/Tangent_plane/crvpbSeG">tangent plane</a>. In <a href="/facts/Inversive_geometry/VqFmbfDI">inversive geometry</a>, a line is a degenerate case of a <a href="/facts/Circle/9fy5ggKn">circle</a>, with infinite radius.</li> <li>Two <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> lines also form a degenerate parabola.</li> <li>A <a href="/facts/Line_segment/V8E0M4qk">line segment</a> can be viewed as a degenerate case of an <a href="/facts/Ellipse/1wUDnGxY">ellipse</a> in which the <a href="/facts/Semiminor_axis/l7JES2wo">semiminor axis</a> goes to zero, the <a href="/facts/Focus_(geometry)/WFtFnqoa">foci</a> go to the endpoints, and the <a href="/facts/Eccentricity_(mathematics)/SzgpW5R7">eccentricity</a> goes to one.</li> <li>A circle can be thought of as a degenerate ellipse, as the <a href="/facts/Eccentricity_(mathematics)/SzgpW5R7">eccentricity</a> approaches 0 and the foci merge.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li> <li>An ellipse can also degenerate into a single point.</li> <li>A <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common <a href="/facts/Asymptote/nvLPuxKa">asymptotes</a>.</li></ul> <h3>Triangle</h3> <p class="note">"Degenerate triangle" redirects here. For the use in computer-graphics meshes, see <a href="/facts/Glossary_of_computer_graphics/GfIomenN">Glossary of computer graphics § degenerate triangles</a>.</p> <p>A degenerate <a href="/facts/Triangle/cRXUwsMn">triangle</a> is a "flat" triangle in the sense that it is contained in a <a href="/facts/Line_segment/V8E0M4qk">line segment</a>. It has thus <a href="/facts/Collinear/SxbVBJYR">collinear</a> vertices<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> and zero area. If the three vertices are all 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 undefined. </p> <h3>Rectangle</h3> <p>A <a href="/facts/Rectangle/mdLfIGKA">rectangle</a> with one pair of opposite sides of length zero degenerates to a line segment, with zero area. If both of the rectangle's pairs of opposite sides have length zero, the rectangle degenerates to a point. </p> <h3>Hyperrectangle</h3> <p>A <a href="/facts/Hyperrectangle/5cbD4khC">hyperrectangle</a> is the n-dimensional analog of a rectangle. If its sides along any of the n axes has length zero, it degenerates to a lower-dimensional hyperrectangle, all the way down to a point if the sides aligned with every axis have length zero. </p> <h3>Convex polygon</h3> <p>A <a href="/facts/Convex_polygon/7x35w610">convex polygon</a> 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 <i>n</i> sides looks like a polygon with fewer sides. In the case of triangles, this definition coincides with the one that has been given above. </p> <h3>Convex polyhedron</h3> <p>A <a href="/facts/Convex_polyhedron/2pRHcRgC">convex polyhedron</a> is degenerate if either two adjacent facets are <a href="/facts/Coplanar/dSTa1bcb">coplanar</a> or two edges are aligned. In the case of a <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a>, this is equivalent to saying that all of its <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a> lie in the same <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a>, giving it a <a href="/facts/Volume/aeAXCBUd">volume</a> of zero. </p> <h3>Standard torus</h3> <ul><li>In contexts where self-intersection is allowed, a double-covered <a href="/facts/Sphere/4Ivb1cTj">sphere</a> is a degenerate <a href="/facts/Standard_torus/J9SkaV1W">standard torus</a> where the axis of revolution passes through the center of the generating circle, rather than outside it.</li> <li>A torus degenerates to a circle when its minor radius goes to 0.</li></ul> <h3>Sphere</h3> <p>When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a <a href="/facts/Point_(geometry)/6YPAvX2a">point</a>. </p> <h3>Other</h3> <p>See <a href="/facts/General_position/p0YpbkJv">general position</a> for other examples. </p> <h2 id="elsewhere">Elsewhere</h2> <ul><li>A set containing a single point is a degenerate <a href="/facts/Linear_continuum/BHs0Wn5m">continuum</a>.</li> <li>Objects such as the <a href="/facts/Digon/vur5IQ5d">digon</a> and <a href="/facts/Monogon/gx05DD40">monogon</a> can be viewed as degenerate cases of <a href="/facts/Polygon/xvkmYhvG">polygons</a>: valid in a general abstract mathematical sense, but not part of the original Euclidean conception of polygons.</li> <li>A <a href="/facts/Random_variable/TwTBXnLT">random variable</a> which can only take one value has a <a href="/facts/Degenerate_distribution/Tj5b5h2z">degenerate distribution</a>; if that value is the real number 0, then its <a href="/facts/Probability_density_function/zvfybna4">probability density</a> is the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>.</li> <li>A <a href="/facts/Root_of_a_function/mlK3dhoT">root</a> of a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> is sometimes said to be <i>degenerate</i> if it is a <a href="/facts/Multiple_root/dndUyetA">multiple root</a>, since generically the <i>n</i> roots of an <i>n</i>th degree polynomial are all distinct.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> This usage carries over to eigenproblems: a degenerate <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> is a multiple root of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a>.</li> <li>In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, any such <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> in the eigenvalues of the <a href="/facts/Hamiltonian_operator/5XwK2HmD">Hamiltonian operator</a> gives rise to <a href="/facts/Degenerate_energy_level/ulrnMXTR">degenerate energy levels</a>. Usually any such degeneracy indicates some underlying <a href="/facts/Symmetry/hpu7DK8p">symmetry</a> in the system.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Degeneracy_(graph_theory)/IaYkus22">Degeneracy (graph theory)</a></li> <li><a href="/facts/Degenerate_form/9TLGBvSH">Degenerate form</a></li> <li><a href="/facts/Trivial_(mathematics)/AqzUxokN">Trivial (mathematics)</a></li> <li><a href="/facts/Pathological_(mathematics)/KmKHjJ1H">Pathological (mathematics)</a></li> <li><a href="/facts/Vacuous_truth/QBsue3pP">Vacuous truth</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. <a href="http://mathworld.wolfram.com/Degenerate.html" target="_blank">http://mathworld.wolfram.com/Degenerate.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Definition of DEGENERACY". www.merriam-webster.com. Retrieved 2019-11-29. <a href="https://www.merriam-webster.com/dictionary/degeneracy" target="_blank">https://www.merriam-webster.com/dictionary/degeneracy</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Mathwords: Degenerate". www.mathwords.com. Retrieved 2019-11-29. <a href="https://www.mathwords.com/d/degenerate.htm" target="_blank">https://www.mathwords.com/d/degenerate.htm</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Mathwords: Degenerate". www.mathwords.com. Retrieved 2019-11-29. <a href="https://www.mathwords.com/d/degenerate.htm" target="_blank">https://www.mathwords.com/d/degenerate.htm</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. <a href="http://mathworld.wolfram.com/Degenerate.html" target="_blank">http://mathworld.wolfram.com/Degenerate.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Mathwords: Degenerate Conic Sections". www.mathwords.com. Retrieved 2019-11-29. <a href="https://www.mathwords.com/d/degenerate_conics.htm" target="_blank">https://www.mathwords.com/d/degenerate_conics.htm</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. <a href="http://mathworld.wolfram.com/Degenerate.html" target="_blank">http://mathworld.wolfram.com/Degenerate.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. <a href="http://mathworld.wolfram.com/Degenerate.html" target="_blank">http://mathworld.wolfram.com/Degenerate.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>"Mathwords: Degenerate". www.mathwords.com. Retrieved 2019-11-29. <a href="https://www.mathwords.com/d/degenerate.htm" target="_blank">https://www.mathwords.com/d/degenerate.htm</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. <a href="http://mathworld.wolfram.com/Degenerate.html" target="_blank">http://mathworld.wolfram.com/Degenerate.html</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>