Unit disk


Unit disk
An open Euclidean unit disk

In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1:

D_1(P) = \{ Q : \vert P-Q\vert<1\}.\,

The closed unit disk around P is the set of points whose distance from P is less than or equal to one:

\bar D_1(P)=\{Q:|P-Q| \leq 1\}.\,

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 \mathbb{D}.

Contents

The open unit disk, the plane, and the upper half-plane

The function

f(z)=\frac{z}{1-|z|^2}

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

g(z)=i\frac{1+z}{1-z}   which is the inverse of the Cayley transform.

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.

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.

Unit disks with respect to other metrics

From top to bottom: open unit disk in the Euclidean metric, taxicab metric, and Chebyshev metric.

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, Stanisław Gołąb 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 or a parallelogram, respectively.

See also

References

  • S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.

External links


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Unit disk graph — In geometric graph theory, a unit disk graph is the intersection graph of a family of unit circles in the Euclidean plane. That is, we form a vertex for each circle, and connect two vertices by an edge whenever the corresponding circles cross… …   Wikipedia

  • Disk (mathematics) — For other uses, see Disc (disambiguation). A disk is the region bounded by a circle. An open disk is the interior of the disk excluding the bounding circle, while a closed disk (see closed set) is the open disk together with the bounding circle.… …   Wikipedia

  • Disk algebra — For other uses, see Disc (disambiguation). In function theory, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions f : D → C, where D is the open unit disk in the complex plane C, f extends to a continuous… …   Wikipedia

  • Unit sphere — some unit spheres In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a …   Wikipedia

  • Unit distance graph — In mathematics, and particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points by an edge whenever the distance between the two points is exactly one.… …   Wikipedia

  • Disk covering problem — The disk covering problem was proposed by C. T. Zahn in 1962. Given an integer n, the problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk. The best solutions to… …   Wikipedia

  • Disk storage — or disc storage is a general category of storage mechanisms, in which data are digitally recorded by various electronic, magnetic, optical, or mechanical methods on a surface layer deposited of one or more planar, round and rotating disks (or… …   Wikipedia

  • Disk II — drives. The Disk II Floppy Disk Subsystem was a 5¼ inch floppy disk drive designed by Steve Wozniak and manufactured by Apple Computer. It was first introduced in 1978 at a retail price of US$495 for pre order; it was later sold for $595… …   Wikipedia

  • Disk encryption theory — Disk encryption is a special case of data at rest protection when the storage media is a sector addressable device (e.g., a hard disk). This article presents cryptographic aspects of the problem. For discussion of different software packages and… …   Wikipedia

  • Disk Wars — is a collectible fantasy tabletop miniature wargame designed primarily by Tom Jolly and Christian T. Petersen and first published in 1999 by Fantasy Flight Games. The game takes its name from its atypical use of cardboard disks to represent… …   Wikipedia