In mathematics, a degenerate case is a limiting or special case of an object class that differs qualitatively and is often simpler. For example, a triangle becomes degenerate when an angle or side length is zero, effectively turning it into a line segment, thus reducing its dimension from two to one, similar to how a circle shrinks to a point. Degeneracies frequently lead to changes in dimension or cardinality, such as the solution set of a system of equations depending on parameters. Though degenerate cases are often linked with singularities, not all special or non-generic cases like right triangles are degenerate. This highlights the concept’s contextual nature in mathematics.
In geometry
Conic section
Main article: Degenerate conic
A degenerate conic is a conic section (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve.
- A point is a degenerate circle, namely one with radius 0.7
- The line is a degenerate case of a parabola if the parabola resides on a tangent plane. In inversive geometry, a line is a degenerate case of a circle, with infinite radius.
- Two parallel lines also form a degenerate parabola.
- A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one.
- A circle can be thought of as a degenerate ellipse, as the eccentricity approaches 0 and the foci merge.8
- An ellipse can also degenerate into a single point.
- A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolae having those lines as common asymptotes.
Triangle
"Degenerate triangle" redirects here. For the use in computer-graphics meshes, see Glossary of computer graphics § degenerate triangles.
A degenerate triangle is a "flat" triangle in the sense that it is contained in a line segment. It has thus collinear vertices9 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.
Rectangle
A rectangle 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.
Hyperrectangle
A hyperrectangle 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.
Convex polygon
A convex polygon 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 n sides looks like a polygon with fewer sides. In the case of triangles, this definition coincides with the one that has been given above.
Convex polyhedron
A convex polyhedron is degenerate if either two adjacent facets are coplanar or two edges are aligned. In the case of a tetrahedron, this is equivalent to saying that all of its vertices lie in the same plane, giving it a volume of zero.
Standard torus
- In contexts where self-intersection is allowed, a double-covered sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it.
- A torus degenerates to a circle when its minor radius goes to 0.
Sphere
When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a point.
Other
See general position for other examples.
Elsewhere
- A set containing a single point is a degenerate continuum.
- Objects such as the digon and monogon can be viewed as degenerate cases of polygons: valid in a general abstract mathematical sense, but not part of the original Euclidean conception of polygons.
- A random variable which can only take one value has a degenerate distribution; if that value is the real number 0, then its probability density is the Dirac delta function.
- A root of a polynomial is sometimes said to be degenerate if it is a multiple root, since generically the n roots of an nth degree polynomial are all distinct.10 This usage carries over to eigenproblems: a degenerate eigenvalue is a multiple root of the characteristic polynomial.
- In quantum mechanics, any such multiplicity in the eigenvalues of the Hamiltonian operator gives rise to degenerate energy levels. Usually any such degeneracy indicates some underlying symmetry in the system.
See also
- Degeneracy (graph theory)
- Degenerate form
- Trivial (mathematics)
- Pathological (mathematics)
- Vacuous truth
References
Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. http://mathworld.wolfram.com/Degenerate.html ↩
"Definition of DEGENERACY". www.merriam-webster.com. Retrieved 2019-11-29. https://www.merriam-webster.com/dictionary/degeneracy ↩
"Mathwords: Degenerate". www.mathwords.com. Retrieved 2019-11-29. https://www.mathwords.com/d/degenerate.htm ↩
"Mathwords: Degenerate". www.mathwords.com. Retrieved 2019-11-29. https://www.mathwords.com/d/degenerate.htm ↩
Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. http://mathworld.wolfram.com/Degenerate.html ↩
"Mathwords: Degenerate Conic Sections". www.mathwords.com. Retrieved 2019-11-29. https://www.mathwords.com/d/degenerate_conics.htm ↩
Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. http://mathworld.wolfram.com/Degenerate.html ↩
Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. http://mathworld.wolfram.com/Degenerate.html ↩
"Mathwords: Degenerate". www.mathwords.com. Retrieved 2019-11-29. https://www.mathwords.com/d/degenerate.htm ↩
Weisstein, Eric W. "Degenerate". mathworld.wolfram.com. Retrieved 2019-11-29. http://mathworld.wolfram.com/Degenerate.html ↩