Unit disk
In
mathematics, the open unit disk around
P , is the set of points whose distance from
P is less than 1:
The closed unit disk around P is the set of points whose distance from P is less than or equal to one:
Unit disks are special cases of disks and
unit balls.
Without further specifications, the term
unit disk is used for the open unit disk about the origin, , with respect to the
standard Euclidean metric. It is the interior of a
circle of radius 1, centered at the origin. This set can be identified with the set of all
complex numbers of
absolute value less than one.
Encyclopedia
In
mathematics, the
open unit disk around
P , is the set of points whose distance from
P is less than 1:
The
closed unit disk around P is the set of points whose distance from P is less than or equal to one:
Unit disks are special cases of disks and
unit balls.
Without further specifications, the term
unit disk is used for the open unit disk about the origin, , with respect to the
standard Euclidean metric. It is the interior of a
circle of radius 1, centered at the origin. This set can be identified with the set of all
complex numbers of
absolute value less than one. When viewed as a subset of the complex plane , the unit disk is often denoted .
The open unit disk, the plane, and the upper half-plane
The function
is an example of a real analytic and
bijective function from the open unit disk to the plane; its inverse function is also analytic. Considered as a real 2-dimensional analytic manifold, the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk is homeomorphic to the whole plane.
There is however no conformal bijective map between the open unit disk and the plane. Considered as a Riemann surface, the open unit disk is therefore different from the
complex plane.
There are conformal bijective maps between the open unit disk and the open upper half-plane. So considered as a Riemann surface, the open unit disk is isomorphic to the upper half-plane, and the two are often used interchangeably.
Much more generally, the Riemann mapping theorem states that every simply connected
open subset of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.
One bijective conformal map from the open unit disk to the open upper half-plane is the
Möbius transformationGeometrically, one can imagine the real axis being bent and shrunk so that the upper half-plane becomes the disk's interior and the real axis forms the disk's circumference, save for one point at the top, the "point at infinity". A bijective conformal map from the open unit disk to the open upper half-plane can also be constructed as the composition of two
stereographic projections: first the unit disk is stereographically projected upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways onto a vertical half-plane touching the sphere, taking the point on the half-sphere opposite to the touching point as projection center.
Topological notions
If considered as
subspaces of the plane with its standard topology, the open unit disk is an
open set and the closed unit disk is a closed set. The boundary of the open or closed unit disk is the
unit circle.
The open unit disk and the closed unit disk are not homeomorphic, since the latter is compact and the former is not. However from the viewpoint of algebraic topology they share many properties: both of them are contractible and so are
homotopy equivalent to a single point. This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to
Z. The
Euler characteristic of a point is 1.
Every continuous map from the closed unit disk to the closed unit disk has at least one fixed point ; this is the case
n=2 of the
Brouwer fixed point theorem. The statement is false for the open unit disk: consider for example
which maps every point of the open unit disk to another point of the open unit disk slightly to the right of the given one.
The one-point compactification of the open unit disk is homeomorphic to a
sphere: imagine the boundary of the open unit disk bent upwards and shrunk, until it meets in one point; this shows that the open unit disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the sphere.
Hyperbolic space
The open unit disk is commonly used as a model for the
hyperbolic plane, by introducing a new metric on it, the
Poincaré metric. Using the above mentioned conformal map between the open unit disk and the upper half-plane, this model can be turned into the Poincaré half-plane model of the hyperbolic plane. Both the Poincaré disk and the Poincaré half-plane are
conformal models of hyperbolic space, i.e. angles measured in the model coincide with angles in hyperbolic space, and consequently the shapes of small figures are preserved.
Another model of hyperbolic space is also built on the open unit disk: the Klein model. It is not conformal, but has the property that straight lines in the model correspond to straight lines in hyperbolic space.
Unit disks with respect to other metrics
One also considers unit disks with respect to other metrics. For instance, with the
taxicab metric and the Chebyshev metric disks look like squares .
The area of the Euclidean unit disk is π and its perimeter is 2π. In contrast, the perimeter of the unit disk in the taxicab geometry is 8. In 1932, Stanislaw Golab proved that in metrics arising from a norm, the perimeter of the unit disk can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disk is a regular
hexagon respectively a
parallelogram.
References
- S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 , 179.
External links
- , by J.C. Álvarez Pavia and A.C. Thompson
