Thespherical coordinate system is commonly used inphysics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distancer, polar angleθ (theta), and azimuthal angleφ (phi). The symbolρ (rho) is often used instead ofr.
Ingeometry, acoordinate system is a system that uses one or morenumbers, orcoordinates, to uniquely determine theposition of thepoints or other geometric elements on amanifold such asEuclidean space.[1][2] The order of the coordinates is significant, and they are sometimes identified by their position in an orderedtuple and sometimes by a letter, as in "thex-coordinate". The coordinates are taken to bereal numbers inelementary mathematics, but may becomplex numbers or elements of a more abstract system such as acommutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers andvice versa; this is the basis ofanalytic geometry.[3]
The simplest example of a coordinate system is the identification of points on aline with real numbers using thenumber line. In this system, an arbitrary pointO (theorigin) is chosen on a given line. The coordinate of a pointP is defined as the signed distance fromO toP, where the signed distance is the distance taken as positive or negative depending on which side of the lineP lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.[4]
The Cartesian coordinate system in three-dimensional space
The prototypical example of a coordinate system is theCartesian coordinate system. In theplane, twoperpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.[5] In three dimensions, three mutuallyorthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes.[6] This can be generalized to createn coordinates for any point inn-dimensional Euclidean space.
Depending on the direction and order of thecoordinate axes, the three-dimensional system may be aright-handed or a left-handed system.
Another common coordinate system for the plane is thepolar coordinate system.[7] A point is chosen as thepole and a ray from this point is taken as thepolar axis. For a given angleθ, there is a single line through the pole whose angle with the polar axis isθ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin isr for given numberr. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example, (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0,θ) for any value ofθ.
There are two common methods for extending the polar coordinate system to three dimensions. In thecylindrical coordinate system, az-coordinate with the same meaning as in Cartesian coordinates is added to ther andθ polar coordinates giving a triple (r, θ, z).[8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ).[9]
A point in the plane may be represented inhomogeneous coordinates by a triple (x, y, z) wherex/z andy/z are the Cartesian coordinates of the point.[10] This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on theprojective plane without the use ofinfinity. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.
Thelog-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
There are ways of describing curves without coordinates, usingintrinsic equations that use invariant quantities such ascurvature andarc length. These include:
Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes,circles orspheres. For example,Plücker coordinates are used to determine the position of a line in space.[11] When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the termline coordinates is used for any coordinate system that specifies the position of a line.
It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to bedualistic. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as theprinciple ofduality.[12]
There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described bycoordinate transformations, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positivex axis, then the coordinate transformation from polar to Cartesian coordinates is given byx = r cosθ andy = r sinθ.
With everybijection from the space to itself two coordinate transformations can be associated:
Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)
For example, in1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more.
"Coordinate line" redirects here; not to be confused withLine coordinates.
Given a coordinate system, if one of the coordinates of a point varies while the other coordinates are held constant, then the resulting curve is called acoordinate curve. If a coordinate curve is astraight line, it is called acoordinate line. A coordinate system for which some coordinate curves are not lines is called acurvilinear coordinate system.[13]Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.
A coordinate line with all other constant coordinates equal to zero is called acoordinate axis, anoriented line used for assigning coordinates. In aCartesian coordinate system, all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwiseorthogonal.
A polar coordinate system is a curvilinear system where coordinate curves are lines orcircles. However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero.
"Coordinate plane" redirects here; not to be confused withPlane coordinates.
Coordinate surfaces of the three-dimensional paraboloidal coordinates.
In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called acoordinate surface. For example, the coordinate surfaces obtained by holdingρ constant in thespherical coordinate system are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak ofcoordinate planes.Similarly,coordinate hypersurfaces are the(n − 1)-dimensional spaces resulting from fixing a single coordinate of ann-dimensional coordinate system.[14]
The concept of acoordinate map, orcoordinate chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is ahomeomorphism from an open subset of a spaceX to an open subset ofRn.[15] It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form anatlas covering the space. A space equipped with such an atlas is called amanifold and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, adifferentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.
Ingeometry andkinematics, coordinate systems are used to describe the (linear) position of points and theangular position of axes, planes, andrigid bodies.[16] In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientationmatrix, which includes, in its three columns, theCartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of threeunit vectors aligned with those axes.
The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of theHellenistic period, a variety of coordinate systems have been developed based on the types above, including:
^Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)".Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01).ISBN978-0-387-18430-2.
