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Unit disk

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Set of points at distance less than one from a given point

For other uses, seeDisc (disambiguation).

Inmathematics, theopen unit disk (ordisc) aroundP (whereP is a given point in theplane), is the set of points whose distance fromP is less than 1:

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

Theclosed unit disk aroundP is the set of points whose distance fromP is less than or equal to one:

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

Unit disks are special cases ofdisks andunit balls; as such, they contain the interior of theunit circle and, in the case of the closed unit disk, the unit circle itself.

Without further specifications, the termunit disk is used for the open unit disk about theorigin, $D_{1}(0)$ , with respect to thestandard Euclidean metric. It is the interior of acircle of radius 1, centered at the origin. This set can be identified with the set of allcomplex numbers ofabsolute value less than one. When viewed as a subset of the complex plane (C), the open unit disk is often denoted $\mathbb {D} =\{z\in \mathbb {C} :|z|<1\}$ . Unlike the notationally similarcircle group $\mathbb {T}$ , the (open) unit disk is not amultiplicative group.

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

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The function

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

is an example of a realanalytic andbijective function from the open unit disk to the plane; its inverse function is also analytic. Considered as a real 2-dimensionalanalytic manifold, the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk ishomeomorphic to the whole plane.

There is however noconformal bijective map between the open unit disk and the plane. Considered as aRiemann surface, the open unit disk is therefore different from thecomplex plane.

There are conformal bijective maps between the open unit disk and the openupper 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, theRiemann mapping theorem states that everysimply 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 theMöbius transformation

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

which is the inverse of theCayley 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 twostereographic 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 forHardy 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 plane

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The open unit disk forms the set of points for thePoincaré disk model of the hyperbolic plane.Circular arcs perpendicular to the unit circle form the "lines" in this model. The unit circle is theCayley absolute that determines ametric on the disk through use ofcross-ratio in the style of theCayley–Klein metric. In the language of differential geometry, the circular arcs perpendicular to the unit circle aregeodesics that show the shortest distance between points in the model. The model includesmotions which are expressed by the special unitary groupSU(1,1). The disk model can be transformed to thePoincaré half-plane model by the mappingg given above.

Both the Poincaré disk and the Poincaré half-plane areconformal models of the hyperbolic plane, which is to say that angles between intersecting curves are preserved by motions of their isometry groups.

Another model of hyperbolic space is also built on the open unit disk: theBeltrami–Klein model. It isnot conformal, but has the property that the geodesics are straight lines.

Unit disks with respect to other metrics

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From top to bottom: open unit disk in theEuclidean metric,taxicab metric, andChebyshev metric.

One also considers unit disks with respect to othermetrics. For instance, with thetaxicab metric and theChebyshev metric disks look like squares (even though the underlyingtopologies are the same as the Euclidean one).

The area of the Euclidean unit disk isπ and itsperimeter 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 anorm, 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 regularhexagon or aparallelogram, respectively.