Editing Unit disk (section)

{{Short description|Set of points at distance less than one from a given point}}
{{other uses|Disc (disambiguation)}}
[[Image:Unit disk open.svg|thumb|right|An open Euclidean unit disk]]
In [[mathematics]], the '''open unit disk''' (or '''disc''') around ''P'' (where ''P'' is a given point in the [[plane (mathematics)|plane]]), is the set of points whose distance from ''P'' is less than 1:

:<math>D_1(P) = \{ Q : \vert P-Q\vert<1\}.\,</math>

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

:<math>\bar D_1(P)=\{Q:|P-Q| \leq 1\}.\,</math>

Unit disks are special cases of [[disk (mathematics)|disks]] and [[unit ball]]s; 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 (mathematics)|origin]], <math>D_1(0)</math>, with respect to the [[Euclidean distance|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 number]]s of [[absolute value]] less than one. When viewed as a subset of the complex plane ('''C'''), the unit disk is often denoted <math>\mathbb{D}</math>.