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
and radius
R is given by the formula
while the closed disk of the same center and radius is given by
The
area of a closed or open disk of radius
R is π
R2 (see
π).
The
ball is the disk generalised to
metric spaces. However, sometimes "disk" is used to mean "ball".
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
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
Category:Euclidean geometryCategory:Rigid bodiesCategory:Articles lacking sources (Erik9bot)ar:قرص (رياضيات)bs:Krugca:Cerclecv:Çавра (геометри)de:Scheibeet:Ringes:Disco (topología)eo:Disko (matematiko)fr:Disque (géométrie)gl:Círculoit:Cerchioko:원판hr:Krugka:წრეწირიlv:Riņķislt:Skritulysnl:Schijf (wiskunde)pl:Kołopt:Círculoru:Кругsk:Kruhsl:Krogsr:Кругfi:Kiekko (matematiikka)sv:Cirkelskivatr:Daire (matematik)uk:Кругvi:Hình trònzh:圆盘
