Editing Disk (mathematics) (section)

==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>