Unit disk

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Unit disk

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the open unit disk around P (where P is a given point in the plane

Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
), 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

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary....
and unit ball

Unit ball

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 fixed central point....
s.

Without further specifications, the term unit disk is used for the open unit disk about the origin

Origin (mathematics)

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 for the geometry of the surrounding space....
, , with respect to the standard Euclidean metric

Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem....
.

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Encyclopedia

In mathematics

Mathematics

Plane (mathematics)

Disk (mathematics)

Unit ball

Origin (mathematics)

Euclidean distance

Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
of radius 1, centered at the origin. This set can be identified with the set of all complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s of absolute value

Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
less than one. When viewed as a subset of the complex plane (C), 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

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions....
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

Analytic manifold

In mathematics, an analytic manifold is a topological manifold with analytic function transition maps. Every complex manifold is an 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

Conformal map

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....
bijective map between the open unit disk and the plane. Considered as a Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold....
, the open unit disk is therefore different from the complex plane

Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished 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

Upper half-plane

In mathematics, the upper half-plane H is the set of complex numberswith positive imaginary part y.The term is associated with a common visualization of complex numbers with points in the plane endowed with Cartesian coordinates, with the Y-axis pointing upwards: the "upper half-plane" corresponds to the half-plane above the X...
. 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

Riemann mapping theorem

In complex analysis, the Riemann mapping theorem states that if is a simply connected space open set of the complex plane which is not all of , then there exists a biholomorphy mapping from onto open unit disk ...
states that every simply connected open subset

Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
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

Möbius transformation

In geometry, a M?bius transformation is a rational function of the form:where z, a, b, c, d are complex numbers satisfying ad − bc ? 0....

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 projection

Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
s: 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

Lebesgue measure

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
while the real line does not.

Topological notions

If considered as subspaces

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology ....
of the plane with its standard topology

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
, the open unit disk is an open set

Open set

Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
. The boundary

Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
of the open or closed unit disk is the unit circle

Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
.

The open unit disk and the closed unit disk are not homeomorphic, since the latter is compact

Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
and the former is not. However from the viewpoint of algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant that classification theorem topological spaces up to homeomorphism....
they share many properties: both of them are contractible

Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map....
and so are homotopy equivalent to a single point. This implies that their fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group or Poincar? group is a group associated to any given pointed space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other....
s are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
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

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. That is to say, x is a fixed point of the function f if and only if f = x....
(we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem

Brouwer fixed point theorem

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 general fixed point theorems....
. 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

Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
: 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.

Hyperbolic space

The open unit disk is commonly used as a model for the hyperbolic plane

Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional quadratic space with a non-singular isotropic quadratic form* A plane in hyperbolic geometry...
, by introducing a new metric on it, the Poincaré metric

Poincaré metric

In mathematics, the Poincar? metric, named after Henri Poincar?, 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

Poincaré half-plane model

In non-Euclidean geometry, the Poincar? half-plane model is the upper half-plane, together with a metric, the Poincar? metric, that makes it a model of two-dimensional hyperbolic geometry....
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

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 geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of the geometry are line segments contained in the disk; that i...
. 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 metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
s. For instance, with the taxicab metric

Taxicab geometry

File:Manhattan distance.svgTaxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric space of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the differences of their coordinates....
and the Chebyshev metric

Chebyshev distance

In mathematics, Chebyshev distance , or Lp space is a Metric defined on a vector space where the distance between two coordinate vectors is the greatest of their differences along any coordinate dimension....
disks look like squares (even though the underlying topologies

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
are the same as the Euclidean one).

The area of the Euclidean unit disk is π and its perimeter

Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
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

Stanislaw Golab

Stanislaw Golab was a Poland mathematician from Krak?w, working in particular on the field of affine geometry.In 1932, he proved that the perimeter of the unit disc can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disc is an affine regular hexagon resp....
proved that in metrics arising from a norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
, 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

Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
respectively a parallelogram

Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
.

External links

, by J.C. Álvarez Pavia and A.C. Thompson

Unit disk

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

Topological notions

Hyperbolic space

Unit disks with respect to other metrics

See also

External links