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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a disk (also spelled disc). is the region in a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
bounded by a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usually denoted as D_r and a closed disk is \overline. However in the field of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
the closed disk is usually denoted as D^2 while the open disk is \operatorname D^2.


Formulas

In
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, the ''open disk'' of center (a, b) and radius ''R'' is given by the formula :D=\ while the ''closed disk'' of the same center and radius is given by :\overline=\. The
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
of a closed or open disk of radius ''R'' is π''R''2 (see
area of a disk In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving this formula, which origi ...
).


Properties

The disk has
circular symmetry In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
. The open disk and the closed disk are not topologically equivalent (that is, they are not
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
), as they have different topological properties from each other. For instance, every closed disk is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
whereas every open disk is not compact. However from the viewpoint of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
they share many properties: both of them are
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
and so are
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to a single point. This implies that their
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s are trivial, and all
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s are trivial except the 0th one, which is isomorphic to Z. The
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of a point (and therefore also that of a closed or open disk) is 1. Every
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
from the closed disk to itself has at least one fixed point (we don't require the map to be
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
or even
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
); this is the case ''n''=2 of the
Brouwer fixed point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...
. The statement is false for the open disk: Consider for example the function f(x,y)=\left(\frac,y\right) 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 x^2 + y^2 = 1 , x >0 .


As a statistical distribution

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 In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
.) "An ingenious argument via elementary functions" shows the mean
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
between two points in the disk to be ,J. S. Lew et al., "On the Average Distances in a Circular Disc" (1977). while direct integration in polar coordinates shows the mean squared distance to be . If we are given an arbitrary location at a distance from the center of the disk, it is also of interest to determine the average distance from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as .


Average distance to an arbitrary internal point

To find we need to look separately at the cases in which the location is internal or external, i.e. in which , and we find that in both cases the result can only be expressed in terms of complete elliptic integrals. If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of under a distribution whose density is for , integrating in polar coordinates centered on the fixed location for which the area of a cell is  ; hence b(q) = \frac \int_0^ \textrm\theta \int_0^ r^2 \textrmr = \frac \int_0^ s(\theta)^3 \textrm\theta. Here can be found in terms of and using the
Law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.; the result is that b(q) = \frac\biggl\ where and are complete elliptic integrals of the first and second kinds. ; .


Average distance to an arbitrary external point

Turning to an external location, we can set up the integral in a similar way, this time obtaining b(q) = \frac \int_0^ \biggl\ \textrm\theta where the law of cosines tells us that and are the roots for of the equation s^2-2qs\,\textrm\theta+q^2\!-\!1=0. Hence b(q) = \frac \int_0^ \biggl\ \textrm\theta. We may substitute to get \beginb(q) &= \frac \int_0^1 \biggl\ \textrmu \\ .6ex&= \frac \int_0^1 \biggl\ \textrmu \\ .6ex&= \frac \biggl\ \\ .6ex&= \frac \biggl\ \end using standard integrals. Hence again , while alsoAbramowitz and Stegun, 17.3.11 et seq. \lim_ b(q) = q + \tfrac.


See also

*
Unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...
, a disk with radius one *
Annulus (mathematics) In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' mean ...
, the region between two concentric circles *
Ball (mathematics) In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
, the usual term for the 3-dimensional analogue of a disk *
Disk algebra In mathematics, specifically in functional and complex analysis, the disk algebra ''A''(D) (also spelled disc algebra) is the set of holomorphic functions :''ƒ'' : D → \mathbb, (where D is the open unit disk in the complex plane \mathbb) th ...
, a space of functions on a disk *
Disk segment In geometry, a circular segment (symbol: ⌓), also known as a disk segment, is a region of a disk (mathematics), disk which is "cut off" from the rest of the disk by a secant line, secant or a chord (geometry), chord. More formally, a circular s ...
*
Orthocentroidal disk In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a subset o ...
, containing certain centers of a triangle


References

{{DEFAULTSORT:Disk (Mathematics) Euclidean geometry Circles Planar surfaces