Unit disk

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the open unit disk (or disc) around P (where P is a given point in the <a href="/facts/Plane_(mathematics)/tAUtRFcn">plane</a>), is the set of points whose distance from P is less than 1:

D
 
 1
 
 
 (
 P
 )
 =
 {
 Q
 :
 |
 P
 −
 Q
 |
 <
 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:

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

Unit disks are special cases of <a href="/facts/Disk_(mathematics)/DgMVr97H">disks</a> and <a href="/facts/Unit_ball/5UeoTcug">unit balls</a>; as such, they contain the interior of the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> 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 <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>, 
 
 
 
 
 D
 
 1
 
 
 (
 0
 )
 
 
 {\displaystyle D_{1}(0)}
 
, with respect to the <a href="/facts/Euclidean_distance/9qDoQKQe">standard Euclidean metric</a>. It is the interior of a <a href="/facts/Circle/9fy5ggKn">circle</a> of radius 1, centered at the origin. This set can be identified with the set of all <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> of <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted 
 
 
 
 
 D
 
 
 
 {\displaystyle \mathbb {D} }
 
.

Unit disk open-in-new

Unit disk