Unit disk

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:

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

The closed unit disk around P is the set of points whose distance from P is less than or equal to one:

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

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, ${\displaystyle D_{1}(0)}$, 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 (C), the unit disk is often denoted ${\displaystyle \mathbb {D} }$.