Unit disk - Citizendia

From top to bottom: open unit disk in the Euclidean metric, taxicab metric, and Chebyshev metric. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry In Mathematics, Chebyshev distance (or Tchebychev distance) or L∞ metric is a metric defined on a Vector space where

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:

$D_1(P) = \{ Q : \vert P-Q\vert<1\}.\,$

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. 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

Without further specifications, the term unit disk is used for the open unit disk about the origin, $D 1 (0)$ , with respect to the standard Euclidean metric. In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler It is the interior of a circle of radius 1, centered at the origin. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the This set can be identified with the set of all complex numbers of absolute value less than one. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. When viewed as a subset of the complex plane (C), the unit disk is often denoted $\mathbb{D}$ .

1 The open unit disk, the plane, and the upper half-plane
2 Topological notions
3 Hyperbolic space
4 Unit disks with respect to other metrics
5 See also
6 References
7 External links

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. This article is about both real and complex analytic functions In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Considered as a real 2-dimensional analytic manifold, the open unit disk is therefore isomorphic to the whole plane. In Mathematics, an analytic manifold is a Topological manifold with analytic transition maps In particular, the open unit disk is homeomorphic to the whole plane. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

There is however no conformal bijective map between the open unit disk and the plane. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane Considered as a Riemann surface, the open unit disk is therefore different from the complex plane. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis

There are conformal bijective maps between the open unit disk and the open upper half-plane. In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} 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. In Complex analysis, the Riemann mapping theorem states that if U is a simply connected open subset of the complex number plane In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

One bijective conformal map from the open unit disk to the open upper half-plane is the Möbius transformation

$g(z)=i\frac{1+z}{1-z}$

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". Möbius transformations should not be confused with the Möbius transform or the Möbius function. 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. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane

The unit disk and the upper half-plane are not interchangeable as domains for Hardy spaces. In Complex analysis, the Hardy spaces (or Hardy classes) H p are certain spaces of holomorphic functions on the unit disk Contributing to this difference is the fact that the unit circle has finite (one-dimensional) Lebesgue measure while the real line does not. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to

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. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open. The boundary of the open or closed unit disk is the unit circle. For a different notion of boundary related to Manifolds see that article In Mathematics, a unit circle is

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. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several The statement is false for the open unit disk: consider for example

$f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)$

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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

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. In In Mathematics, the Poincaré metric, named after Henri Poincaré, is the Metric tensor describing a two-dimensional surface of constant negative Curvature 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. In Non-Euclidean geometry, the Poincaré half-plane model is the Upper half-plane, together with a metric the Poincaré metric, that makes it a model 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. In geometry the Klein model, also called the projective model the Beltrami–Klein model the Klein–Beltrami model and the Cayley–Klein model is a model of n-dimensional Hyperbolic 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. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. 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). Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry In Mathematics, Chebyshev distance (or Tchebychev distance) or L∞ metric is a metric defined on a Vector space where Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

The area of the Euclidean unit disk is π and its perimeter is 2π. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The perimeter is the distance around a given two-dimensional object 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. Year 1932 ( MCMXXXII) was a Leap year starting on Friday of the Gregorian calendar. Stanisław Gołąb ( July 26 1902 – April 30 1980) was a Polish Mathematician from Kraków, working in particular In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides

References

S. In Geometric graph theory, a unit disk graph is the Intersection graph of a family of Unit circles in the Euclidean plane. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 179.

External links

Eric W. Weisstein, Unit disk at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
On the Perimeter and Area of the Unit Disc, by J. C. Álvarez Pavia and A. C. Thompson

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents