Conic section 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. • 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. • 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 A degenerate
triangle is a "flat" triangle in the sense that it is contained in a
line segment. It has thus
collinear vertices 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 -dimensional analog of a rectangle. If its sides along any of the 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==