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: Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ...

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. In geometry, a disk is the region in a plane contained inside of a circle. ... 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. ...

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 (), the unit disk is often denoted . In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ... In mathematics the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

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. In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an analytic manifold is a topological manifold with analytic transition maps. ... This word should not be confused with homomorphism. ...

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. In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

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. In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...

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. The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

One bijective conformal map from the open unit disk to the open upper half plane is the Möbius transformation In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

which maps 1 to 1, -i to 0, -1 to -1, and i to ∞. 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. Stereographic projection of a circle of radius R onto the x axis. ...

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. 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 (or the relative topology, or the induced topology). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ... Illustration of a 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. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... 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 homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

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 In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily 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, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D n to itself has a fixed point. ...

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. In mathematics, compactification is applied to topological spaces to make them compact spaces. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ...

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. A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In mathematics, the Poincaré metric is the natural metric tensor for Poincaré half-plane model of hyperbolic geometry. ... In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ...

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. 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...

Unit disks with respect to other metrics

Examples of unit disks with respect to different 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). Image File history File links Unit_disc. ... Image File history File links Unit_disc. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... Manhattan versus Euclidean distance: The red, blue, and yellow lines all have the same length (12), whereas the green line has length . ... In a plane, the Chebyshev distance between the point P1 with coordinates (x1, y1) and the point P2 at (x2, y2) is This concept is named after Pafnuty Chebyshev. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

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. Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... The perimeter is the distance around a given two-dimensional object. ... 1932 (MCMXXXII) was a leap year starting on Friday (the link will take you to a full 1932 calendar). ... StanisÅ‚aw GoÅ‚Ä…b was a 20th century Polish mathematician from Kraków, working in particular on the field of affine geometry. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... A regular hexagon In geometry, a hexagon is a polygon with six edges and six vertices. ... A parallelogram. ...

References

• S. Golab, "Quelques problèmes métriques de la géometrie de Minkowski", Trav. de l'Acad. Mines Cracovie 6 (1932), 1­79.

External links

• On the Perimeter and Area of the Unit Disc, by J.C. Álvarez Pavia and A.C. Thompson