Disk (mathematics, $D_{R}$ )

A planar Euclidean disk consists of the points within a fixed radius of a centre, with either the open or closed boundary convention. The closed disk centered at the origin is

D_{R} = {x \in R^{2} : ∣ x ∣ \leq R}

and has area

π R^{2}

. It is the two-dimensional closed Euclidean ball; the general closed disc may have any dimension.

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Semicircular disk ( ${(x, y) : x^{2} + y^{2} \leq a^{2}, y \geq 0}$ )

 0  0

A semicircular disk is one half of a disk cut along a diameter, including its interior. It is a circular sector of angle

π

, with area

π a^{2} /2

. Source polar coordinates use

0 \leq r \leq a

0 \leq θ \leq π

, and area element

r d r d θ

. Its boundary consists of a semicircle and a diameter; the semicircle alone is a curve and has no planar interior.

Open disk ( $B^{2} = {v \in R^{2} : ∣ v ∣ < 1}$ )

 0  0

The open disk is the interior of a planar disk. Any such open disk is homeomorphic to

R^{2}

, and in particular is contractible and simply connected.

Lens-shaped intersection of two disks ( $A = 2 r^{2} arccos [d / (2 r)] - (d /2) 4 r^{2} - d^{2}$ )

 0  0

For two disks of equal radius

r

whose centres are separated by

0 \leq d \leq 2 r

, their intersection consists of two circular segments. Each segment is a sector of angle

2 θ

minus an isosceles triangle, where

cos θ = d / (2 r)

. Hence its total area is

A = 2 r^{2} θ - r^{2} sin (2 θ) = 2 r^{2} arccos \frac{d}{2 r} - \frac{d}{2} 4 r^{2} - d^{2} .

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The limiting cases are

A = π r^{2}

d = 0

and

A = 0

d = 2 r

. This geometric region is distinct from an optical lens.

Open disc

 0  0

The open disc with centre

z

and radius

r > 0

{w : ∣ w - z ∣ < r}

Unit disc ( $D$ )

 0  0

The unit disc in the complex plane is

D = {z \in C : ∣ z ∣ < 1}

Automorphism of the unit disk ( $ϕ (z) = e^{i θ} \frac{z - a}{1 - a z}$ )

 0  0

A biholomorphism of the unit disk to itself. Every such map has the form

e^{i θ} (z - a) / (1 - \overline{a} z)

with

∣ a ∣ < 1

. These maps normalize an extremal univalent function in the proof of the Riemann mapping theorem.

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In mathematics, particularly in geometry, a "disk" (or "disc") refers to a two-dimensional shape that is defined as the region in the plane that is enclosed by a circle. The term can have slightly different meanings depending on the context: 1. **Closed Disk**: This includes all the points inside a circle as well as the points on the boundary (the circumference of the circle).

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Disk (mathematics, $D_{R}$ )

Semicircular disk ( ${(x, y) : x^{2} + y^{2} \leq a^{2}, y \geq 0}$ )

Open disk ( $B^{2} = {v \in R^{2} : ∣ v ∣ < 1}$ )

Lens-shaped intersection of two disks ( $A = 2 r^{2} arccos [d / (2 r)] - (d /2) 4 r^{2} - d^{2}$ )

Open disc

Unit disc ( $D$ )

Automorphism of the unit disk ( $ϕ (z) = e^{i θ} \frac{z - a}{1 - a z}$ )

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