Disk (mathematics)


Disk (mathematics)
A disk is the region bounded by a circle. An open disk is the interior of the disk excluding the bounding circle, while a closed disk (see closed set) is the open disk together with the bounding circle.

In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle.

A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary. In Cartesian coordinates, the open disk of center (a,b) and radius R is given by the formula

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

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

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

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

The ball is the disk generalised to metric spaces. In context, the term ball may be used instead of disk.

In theoretical physics a disk is a rigid body which is capable of participating in collisions in a two-dimensional gas. Usually the disk is considered rigid so that collisions are deemed elastic.

Geometry

The Euclidean disk is circular symmetrical.

Topological notions

The open disk and the closed disk are not homeomorphic, since the latter is compact and the former is not. 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. This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.

Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem. The statement is false for the open disk: consider for example

f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)

which maps every point of the open unit disk to another point of the open unit disk slightly to the right of the given one.

See also


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