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| [[File:Circle-withsegments.svg|thumb|200px|right|Disc with [[circumference]] (C) in black, [[diameter]] (D) in cyan, [[radius]] (R) in red, and [[centre (geometry)|centre]] (O) in magenta.]]
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| In [[geometry]], a '''disk''' (also [[Spelling of disc|spelled]] '''disc''') is the region in a [[plane (geometry)|plane]] bounded by a [[circle]]. | |
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| 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 <math>(a, b)</math> and radius ''R'' is given by the formula
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| :<math>D=\{(x, y)\in {\mathbb R^2}: (x-a)^2+(y-b)^2 < R^2\}</math>
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| while the closed disk of the same center and radius is given by
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| :<math>\overline{ D }=\{(x, y)\in {\mathbb R^2}: (x-a)^2+(y-b)^2 \le R^2\}.</math>
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| The [[area (geometry)|area]] of a closed or open disk of radius ''R'' is π''R''<sup>2</sup> (see [[area of a disk]]).
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| The ''[[Ball (mathematics)|ball]]'' is the disk generalised to [[metric spaces]]. In context, the term ''ball'' may be used instead of ''disk''.
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| 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 collision|elastic]].
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| ==Geometry==
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| The Euclidean disk is [[Circular symmetry|circular symmetrical]].
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| ==Topological notions==
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| The open disk and the closed disk are not homeomorphic, since the latter is [[compact space|compact]] and the former is not. However from the viewpoint of [[algebraic topology]] they share many properties: both of them are [[contractible space|contractible]] and so are [[homotopy equivalent]] to a single point. This implies that their [[fundamental group]]s are trivial, and all [[homology group]]s 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.
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| Every [[continuous map]] from the closed disk to itself has at least one [[fixed point (mathematics)|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
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| :<math>f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)</math> | |
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| 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.
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| ==See also==
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| *[[Unit disk]], a disk with radius one
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| *[[Annulus (mathematics)]]
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| *[[Ball (mathematics)]], the usual term for the 3-dimensional disk
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| *[[Disk algebra]]
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| *[[Lentoid]]
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| *[[Moment of inertia of a uniform disc]]
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| {{DEFAULTSORT:Disk (Mathematics)}}
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| [[Category:Euclidean geometry]]
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| [[Category:Rigid bodies]]
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