Disk (mathematics)

In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not.

For a radius

r

{\displaystyle r}

, an open disk is usually denoted as

D

r

{\displaystyle D_{r}}

, and a closed disk is

D

r

¯

{\displaystyle {\overline {D_{r}}}}

. However in the field of topology the closed disk is usually denoted as

D

2

{\displaystyle D^{2}}

, while the open disk is

int

⁡

D

2

{\displaystyle \operatorname {int} D^{2}}

.

Formulas

In Cartesian coordinates, the open disk with center

(

a

,

b

)

{\displaystyle (a,b)}

and radius R is given by the formula

D

=

{

(

x

,

y

)

∈

R

2

:

(

x

−

a

)

2

+

(

y

−

b

)

2

<

R

2

}

,

{\displaystyle D=\{(x,y)\in \mathbb {R} ^{2}:(x-a)^{2}+(y-b)^{2}<R^{2}\},}

while the closed disk with the same center and radius is given by

D

¯

=

{

(

x

,

y

)

∈

R

2

:

(

x

−

a

)

2

+

(

y

−

b

)

2

≤

R

2

}

.

{\displaystyle {\overline {D}}=\{(x,y)\in \mathbb {R} ^{2}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.}

The area of a closed or open disk of radius R is πR2 (see area of a disk).

Properties

The disk has circular symmetry.

The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact. However from the viewpoint of algebraic topology they share many properties: both of them are contractible and so are homotopy equivalent to a single point.