Unit disk
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In mathematics, the open unit disk around P (where P is a given point in the plane), 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, D1(0), 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 (C), the unit disk is often denoted 
.
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[edit] 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 ("biholomorphic", or "conformally equivalent") 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 transformation
Geometrically, 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.
The unit disk and the upper half-plane are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not.
[edit] 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 (and therefore also that of a closed or open disk) is 1.
Every continuous map from the closed unit disk to the closed unit disk has at least one fixed point (we don't require the map to be bijective or even surjective); 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 (compact) sphere.
[edit] 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 (but not the sizes) 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.
[edit] 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 (even though the underlying topologies are the same as the Euclidean one).
The area of the Euclidean unit disk is π and its perimeter is 2π. In contrast, the perimeter (relative to the taxicab metric) 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.
[edit] See also
[edit] References
- S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.
[edit] External links
- Eric W. Weisstein, Unit disk at MathWorld.
- On the Perimeter and Area of the Unit Disc, by J.C. Álvarez Pavia and A.C. Thompson
