Editing Disk (mathematics)

{{Short description|Plane figure, bounded by circle}}
{{Redirect|2-ball|the basketball event|2Ball}}
{{Other uses|Disc (disambiguation){{!}}Disc}}

[[File:Circle-withsegments.svg|thumb|right|Disk with {{legend-line|black solid 3px|[[circumference]] ''C''}}
{{legend-line|blue solid 2px|diameter ''D''}}
{{legend-line|red solid 2px|radius ''R''}}
{{legend-line|green solid 2px|center or origin ''O''}}]]

In [[geometry]], a '''disk''' ([[Spelling of disc|also spelled]] '''disc''')<ref name="odm">{{cite book |title=The Concise Oxford Dictionary of Mathematics |first1=Christopher |last1=Clapham |first2=James |last2=Nicholson |publisher=Oxford University Press |year=2014 |isbn=9780199679591 |url=https://books.google.com/books?id=c69GBAAAQBAJ&pg=PA138 |page=138}}</ref> is the region in a [[plane (geometry)|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.<ref>{{cite book |title=Intuitive Concepts in Elementary Topology |series=Dover Books on Mathematics |first=B. H. |last=Arnold |publisher=Courier Dover Publications |year=2013 |isbn=9780486275765 |page=58 |url=https://books.google.com/books?id=TsbDAgAAQBAJ&pg=PA58}}</ref>

For a radius <math>r</math>, an open disk is usually denoted as <math>D_r</math>, and a closed disk is <math>\overline{D_r}</math>. However in the field of [[topology]] the closed disk is usually denoted as <math>D^2</math>, while the open disk is <math>\operatorname{int} D^2</math>.

==Formulas==
In [[Cartesian coordinates]], the ''open disk'' with center <math>(a, b)</math> and radius ''R'' is given by the formula<ref name="odm"/>
<math display="block">
 D = \{(x, y) \in \mathbb{R}^2 : (x - a)^2 + (y - b)^2 < R^2\},
</math>
while the ''closed disk'' with the same center and radius is given by
<math display="block">
 \overline{D} = \{(x, y) \in \mathbb{R}^2 : (x - a)^2 + (y - b)^2 \le R^2\}.
</math>

The [[area (geometry)|area]] of a closed or open disk of radius ''R'' is π''R''<sup>2</sup> (see [[area of a disk]]).<ref>{{cite book |title=Journey into Mathematics: An Introduction to Proofs |series=Dover Books on Mathematics |first=Joseph J. |last=Rotman |publisher=Courier Dover Publications |year=2013 |isbn=9780486151687 |page=44 |url=https://books.google.com/books?id=6QDCAgAAQBAJ&pg=PA44-IA52}}.</ref>

==Properties==
The disk has [[circular symmetry]].<ref>{{Cite book|url=https://archive.org/details/iconssymmetries0000altm|url-access=registration|quote=disc circular symmetry.|title=Icons and Symmetries|last=Altmann|first=Simon L.|date=1992|publisher=Oxford University Press|isbn=9780198555995|language=en}}</ref>

The open disk and the closed disk are not topologically equivalent (that is, they are not [[homeomorphism|homeomorphic]]), as they have different topological properties from each other. For instance, every closed disk is [[compact space|compact]] whereas every open disk is not compact.<ref>{{citation|title=New Foundations for Physical Geometry: The Theory of Linear Structures|first=Tim|last=Maudlin|publisher=Oxford University Press|year=2014|isbn=9780191004551|page=339|url=https://books.google.com/books?id=kEbbAgAAQBAJ&pg=PA339}}.</ref> However from the viewpoint of [[algebraic topology]] they share many properties: both of them are [[contractible space|contractible]]<ref>{{citation|title=Combinatorial Group Theory: A Topological Approach|volume=14|series=London Mathematical Society Student Texts|first=Daniel E.|last=Cohen|publisher=Cambridge University Press|year=1989|isbn=9780521349369|url=https://books.google.com/books?id=STc4AAAAIAAJ&pg=PA79|page=79}}.</ref> 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.<ref>In higher dimensions, the Euler characteristic of a closed ball remains equal to +1, but the Euler characteristic of an open ball is +1 for even-dimensional balls and &minus;1 for odd-dimensional balls. See {{citation|title=Introduction to Geometric Probability|first1=Daniel A.|last1=Klain|first2=Gian-Carlo|last2=Rota|author2-link=Gian-Carlo Rota|pages=46–50|publisher=Cambridge University Press|series=Lezioni Lincee|year=1997}}.</ref>

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]].<ref>{{harvtxt|Arnold|2013}}, p.&nbsp;132.</ref> The statement is false for the open disk:<ref>{{harvtxt|Arnold|2013}}, Ex.&nbsp;1, p.&nbsp;135.</ref>

Consider for example the function
<math>f(x,y)=\left(\frac{x+\sqrt{1-y^2}}{2},y\right)</math>
which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle <math>x^2 + y^2 = 1 , x >0 .</math>

==As a statistical distribution==
[[File:Discdist.svg|thumb|The average distance to a location from points on a disc]]A uniform distribution on a unit circular disk is occasionally encountered in statistics. It most commonly occurs in operations research in the mathematics of urban planning, where it may be used to model a population within a city. Other uses may take advantage of the fact that it is a distribution for which it is easy to compute the probability that a given set of linear inequalities will be satisfied. ([[Multivariate normal distribution|Gaussian distributions]] in the plane require [[Numerical integration|numerical quadrature]].)

"An ingenious argument via elementary functions" shows the mean [[Euclidean distance]] between two points in the disk to be {{math|{{sfrac|128|45π}} ≈ 0.90541}},<ref name=lew>J. S. Lew et al., "On the Average Distances in a Circular Disc" (1977).</ref> while direct integration in polar coordinates shows the mean squared distance to be {{math|1}}.

If we are given an arbitrary location at a distance {{math|''q''}} from the center of the disk, it is also of interest to determine the average distance {{math|''b''(''q'')}} from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as {{math|''q''<sup>2</sup>+{{sfrac|1|2}}}}.

===Average distance to an arbitrary internal point===
[[File:Cjcdiscin.svg|thumb|The average distance from a disk to an internal point]]To find {{math|''b''(''q'')}} we need to look separately at the cases in which the location is internal or external, i.e. in which {{math|''q'' ≶ 1}}, and we find that in both cases the result can only be expressed in terms of [[elliptic integrals|complete elliptic integrals]].

If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of {{math|''r''}} under a distribution whose density is {{math|{{sfrac|1|π}}}} for {{math|0 &le; ''r'' &le; ''s''(θ)}}, integrating in polar coordinates centered on the fixed location for which the area of a cell is {{math|''r'' d''r'' dθ}}&nbsp;; hence
<math display="block">b(q) = \frac{1}{\pi} \int_0^{2\pi} \textrm{d}\theta \int_0^{s(\theta)} r^2 \textrm{d}r  = \frac{1}{3\pi} \int_0^{2\pi} s(\theta)^3 \textrm{d}\theta.</math>

Here {{math|''s''(θ)}} can be found in terms of {{math|''q''}} and {{math|θ}} using the [[Law of cosines]]. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.;<ref name=lew/> the result is that
<math display="block">b(q) = \frac{4}{9\pi}\biggl\{ 4(q^2-1)K(q^2) + (q^2+7)E(q^2)\biggr\} </math>
where {{math|''K''}} and {{math|''E''}} are complete elliptic integrals of the first and second kinds.<ref>[[Abramowitz and Stegun]], 17.3.</ref> {{math|''b''(0) {{=}} {{sfrac|2|3}}}}; {{math|''b''(1) {{=}} {{sfrac|32|9π}} ≈ 1.13177}}.

===Average distance to an arbitrary external point===
[[File:Cjcdiscex.svg|thumb|The average distance from a disk to an external point]]Turning to an external location, we can set up the integral in a similar way, this time obtaining
<math display="block">b(q) = \frac{2}{3\pi} \int_0^{\textrm{sin}^{-1}\tfrac{1}{q}} \biggl\{ s_{+}(\theta)^3-s_{-}(\theta)^3\biggr\} \textrm{d}\theta</math>
where the law of cosines tells us that {{math|''s''<sub>+</sub>(θ)}} and {{math|''s''<sub>–</sub>(θ)}} are the roots for {{math|''s''}} of the equation
<math display="block">s^2-2qs\,\textrm{cos}\theta+q^2\!-\!1=0.</math>
Hence
<math display="block">b(q) = \frac{4}{3\pi} \int_0^{\textrm{sin}^{-1}\tfrac{1}{q}} \biggl\{ 3q^2\textrm{cos}^2\theta \sqrt{1-q^2 \textrm{sin}^2\theta} +  \Bigl( 1-q^2 \textrm{sin}^2\theta\Bigr)^{\tfrac{3}{2}} \biggl\} \textrm{d}\theta.
</math>
We may substitute {{math|''u'' {{=}} ''q'' sinθ }} to get 
<math display="block">\begin{align}b(q) &= \frac{4}{3\pi} \int_0^1 \biggl\{ 3\sqrt{q^2-u^2} \sqrt{1-u^2} + \frac{(1-u^2)^{\tfrac{3}{2}}}{\sqrt{q^2-u^2}} \biggr\} \textrm{d}u \\[0.6ex] &=
\frac{4}{3\pi} \int_0^1 \biggl\{ 4\sqrt{q^2-u^2} \sqrt{1-u^2} - \frac{q^2-1}{q} \frac{\sqrt{1-u^2}}{\sqrt{q^2-u^2}} \biggr\} \textrm{d}u \\[0.6ex] &=
\frac{4}{3\pi}  \biggl\{ \frac{4q}{3} \biggl( (q^2+1)E(\tfrac{1}{q^2})-(q^2-1)K(\tfrac{1}{q^2}) \biggr) - (q^2-1) \biggl(qE(\tfrac{1}{q^2})-\frac{q^2-1}{q}K(\tfrac{1}{q^2}) \biggr) \biggr\} \\[0.6ex] &=
\frac{4}{9\pi}  \biggl\{ q(q^2+7)E(\tfrac{1}{q^2}) - \frac{q^2-1}{q}(q^2+3)K(\tfrac{1}{q^2}) \biggr\} 
\end{align}</math>
using standard integrals.<ref>[[Gradshteyn and Ryzhik]] 3.155.7 and 3.169.9, taking due account of the difference in notation from Abramowitz and Stegun. (Compare A&amp;S 17.3.11 with G&amp;R 8.113.) This article follows A&amp;S's notation.</ref>

Hence again {{math|''b''(1) {{=}} {{sfrac|32|9π}}}}, while also<ref>Abramowitz and Stegun, 17.3.11 et seq.</ref>
<math display="block">\lim_{q \to \infty} b(q) = q + \tfrac{1}{8q}.</math>

==See also==
*[[Unit disk]], a disk with radius one
*[[Annulus (mathematics)]], the region between two concentric circles
*[[Ball (mathematics)]], the usual term for the 3-dimensional analogue of a disk
*[[Disk algebra]], a space of functions on a disk
*[[Circular segment]]
*[[Orthocentroidal disk]], containing certain centers of a triangle

==References==
{{Reflist|30em}}

{{Compact topological surfaces}}
{{Authority control}}

{{DEFAULTSORT:Disk (Mathematics)}}
[[Category:Euclidean geometry]]
[[Category:Circles]]
[[Category:Planar surfaces]]