Бірлік дискісі және оның қасиеттері

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

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:

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; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term unit disk is used for the open unit disk about the origin, , 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 .

The open unit disk forms the set of points for the Poincaré disk model of the hyperbolic plane. Circular arcs perpendicular to the unit circle form the "lines" in this model. The unit circle is the Cayley absolute that determines a metric on the disk through use of cross ratio in the style of the Cayley–Klein metric. In the language of differential geometry, the circular arcs perpendicular to the unit circle are geodesics that show the shortest distance between points in the model. The model includes motions which are expressed by the special unitary group SU(1,1). The disk model can be transformed to the Poincaré half plane model by the mapping g given above. Both the Poincaré disk and the Poincaré half plane are conformal 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: the Beltrami Klein model. It is not conformal, but has the property that the geodesics are straight lines.

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.