Disk (mathematics)

Source: Wikipedia, the free encyclopedia.
Disk with
  diameter D
  radius R
  center or origin O

In

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.[2]

For a radius, , an open disk is usually denoted as and a closed disk is . However in the field of topology the closed disk is usually denoted as while the open disk is .

Formulas

In

Cartesian coordinates
, the open disk of center and radius R is given by the formula:[1]

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

The

area of a disk).[3]

Properties

The disk has circular symmetry.[4]

The open disk and the closed disk are not topologically equivalent (that is, they are not

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.[7]

Every

Brouwer fixed point theorem.[8] The statement is false for the open disk:[9]

Consider for example the function 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

As a statistical distribution

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. (Gaussian distributions in the plane require numerical quadrature.)

"An ingenious argument via elementary functions" shows the mean Euclidean distance between two points in the disk to be 128/45π ≈ 0.90541,[10] while direct integration in polar coordinates shows the mean squared distance to be 1.

If we are given an arbitrary location at a distance q from the center of the disk, it is also of interest to determine the average distance 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 q2+1/2.

Average distance to an arbitrary internal point

complete elliptic integrals
.

If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of r under a distribution whose density is 1/π for 0 ≤ rs(θ), integrating in polar coordinates centered on the fixed location for which the area of a cell is r dr ; hence

Here s(θ) can be found in terms of q and 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.;[10] the result is that

where K and E are complete elliptic integrals of the first and second kinds.[11] b(0) = 2/3; b(1) = 32/ ≈ 1.13177.

Average distance to an arbitrary external point

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

where the law of cosines tells us that s+(θ) and s(θ) are the roots for s of the equation
Hence
We may substitute u = q sinθ to get
using standard integrals.[12]

Hence again b(1) = 32/, while also[13]

See also

References

  1. ^ .
  2. .
  3. .
  4. . disc circular symmetry.
  5. .
  6. .
  7. ^ 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 −1 for odd-dimensional balls. See Klain, Daniel A.; Rota, Gian-Carlo (1997), Introduction to Geometric Probability, Lezioni Lincee, Cambridge University Press, pp. 46–50.
  8. ^ Arnold (2013), p. 132.
  9. ^ Arnold (2013), Ex. 1, p. 135.
  10. ^ a b J. S. Lew et al., "On the Average Distances in a Circular Disc" (1977).
  11. ^ Abramowitz and Stegun, 17.3.
  12. ^ Gradshteyn and Ryzhik 3.155.7 and 3.169.9, taking due account of the difference in notation from Abramowitz and Stegun. (Compare A&S 17.3.11 with G&R 8.113.) This article follows A&S's notation.
  13. ^ Abramowitz and Stegun, 17.3.11 et seq.