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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 coordinate system is a system that uses one or more
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s, or coordinates, to uniquely determine the position of the
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...
or other geometric elements on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
such as
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
and sometimes by a letter, as in "the ''x''-coordinate". The coordinates are taken to be
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s in elementary mathematics, but may be
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s or elements of a more abstract system such as a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
.


Common coordinate systems


Number line

The simplest example of a coordinate system is the identification of points on a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of a point ''P'' is defined as the signed distance from ''O'' to ''P'', where the signed distance is the distance taken as positive or negative depending on which side of the line ''P'' lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.


Cartesian coordinate system

The prototypical example of a coordinate system is the
Cartesian coordinate system 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 ...
. In the plane, two
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create ''n'' coordinates for any point in ''n''-dimensional Euclidean space. Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left-handed system. This is one of many coordinate systems.


Polar coordinate system

Another common coordinate system for the plane is the ''polar coordinate system''. A point is chosen as the ''pole'' and a ray from this point is taken as the ''polar axis''. For a given angle ''θ'', there is a single line through the pole whose angle with the polar axis is ''θ'' (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is ''r'' for given number ''r''. 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''+2''π'') and (−''r'', ''θ''+''π'') are all polar coordinates for the same point. The pole is represented by (0, ''θ'') for any value of ''θ''.


Cylindrical and spherical coordinate systems

There are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a ''z''-coordinate with the same meaning as in Cartesian coordinates is added to the ''r'' and ''θ'' polar coordinates giving a triple (''r'', ''θ'', ''z''). 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'').


Homogeneous coordinate system

A point in the plane may be represented in ''homogeneous coordinates'' by a triple (''x'', ''y'', ''z'') where ''x''/''z'' and ''y''/''z'' are the Cartesian coordinates of the point. 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 the projective plane without the use of
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.


Other commonly used systems

Some other common coordinate systems are the following: * Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves. ** Orthogonal coordinates: coordinate surfaces meet at right angles **
Skew coordinates A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates si ...
: coordinate surfaces are not orthogonal * The log-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. *
Plücker coordinates In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, P3. Because they satisfy a quadratic constraint, they establish a one-to- ...
are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
. * Generalized coordinates are used in the Lagrangian treatment of mechanics. *
Canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
are used in the Hamiltonian treatment of mechanics. *
Barycentric coordinate system In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The ...
as used for
ternary plot A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an eq ...
s and more generally in the analysis of
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s. *
Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
are used in the context of triangles. There are ways of describing curves without coordinates, using
intrinsic equation In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. The ...
s that use invariant quantities such as
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
and
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
. These include: * The
Whewell equation The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tangential angle is the angle between the tangent to the curve and the -axis, and the arc length is the distance along the cu ...
relates arc length and the
tangential angle In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of ...
. * The Cesàro equation relates arc length and curvature.


Coordinates of geometric objects

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,
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 cons ...
s or
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s. For example,
Plücker coordinates In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, P3. Because they satisfy a quadratic constraint, they establish a one-to- ...
are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term '' line 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 be ''dualistic''. 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 the ''principle of duality''.


Transformations

There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by ''coordinate 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 positive ''x'' axis, then the coordinate transformation from polar to Cartesian coordinates is given by ''x'' = ''r'' cos''θ'' and ''y'' = ''r'' sin''θ''. With every
bijection 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 ...
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, in 1D, 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 lines/curves and planes/surfaces

In two dimensions, if one of the coordinates in a point coordinate system is held constant and the other coordinate is allowed to vary, then the resulting curve is called a coordinate curve. If the coordinate curves are, in fact,
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
s, they may be called coordinate lines. In Cartesian coordinate systems, coordinates lines are mutually orthogonal, and are known as ''
coordinate axes 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 ...
''. For other coordinate systems the coordinates curves may be general curves. For example, the coordinate curves in polar coordinates obtained by holding ''r'' constant are the circles with center at the origin. A coordinate system for which some coordinate curves are not lines is called a ''
curvilinear coordinate system In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
''. This procedure does not always make sense, for example there are no coordinate curves in a homogeneous coordinate system. In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a coordinate surface. For example, the coordinate surfaces obtained by holding ''ρ'' constant in the
spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
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 of coordinate planes. Similarly, coordinate hypersurfaces are the -dimensional spaces resulting from fixing a single coordinate of an ''n''-dimensional coordinate system.


Coordinate maps

The concept of a ''coordinate map'', or ''coordinate 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 a
homeomorphism 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 isom ...
from an open subset of a space ''X'' to an open subset of R''n''. 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 an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
covering the space. A space equipped with such an atlas is called a ''manifold'' and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.


Orientation-based coordinates

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 ...
and
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
, coordinate systems are used to describe the (linear) position of points and the
angular position In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imag ...
of axes, planes, and rigid bodies. 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 orientation
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, which includes, in its three columns, the
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 ...
of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s aligned with those axes.


Geographic systems

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 the
Hellenistic period In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
, a variety of coordinate systems have been developed based on the types above, including: *
Geographic coordinate system The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...
, the spherical coordinates of
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
* Projected coordinate systems, including thousands of
cartesian coordinate system 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 ...
s, each based on a
map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and l ...
to create a planar surface of the world or a region. * Geocentric coordinate system, a three-dimensional
cartesian coordinate system 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 ...
that models the earth as an object, and are most commonly used for modeling the orbits of
satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioiso ...
s, including the
Global Positioning System The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite ...
and other
satellite navigation A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows satellite navigation devices to determine their location ( longitude, latitude, and altitude/ elevation) to hig ...
systems.


See also

*
Absolute angular momentum In meteorology, absolute angular momentum refers to the angular momentum in an 'absolute' coordinate system (absolute time and space). Introduction Angular momentum equates with the cross product of the position (vector) of a particle (or flu ...
* Alphanumeric grid * Axes conventions in engineering * Celestial coordinate system * Coordinate-free * Fractional coordinates *
Frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
*
Galilean transformation In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotat ...
* Grid reference * Nomogram, graphical representations of different coordinate systems *
Reference system In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
*
Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
*
Translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...


Relativistic coordinate systems

*
Eddington–Finkelstein coordinates In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (e.g. a spherically symmetric black hole) which are adapted to radial null geodesics. Null geodesics are the worldlines of pho ...
* Gaussian polar coordinates *
Gullstrand–Painlevé coordinates Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are such that the time coordinate follows the pr ...
* Isotropic coordinates *
Kruskal–Szekeres coordinates In general relativity, Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacet ...
*
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coor ...


References


Citations


Sources

* * *


External links


Hexagonal Coordinate Systems
{{Authority control Analytic geometry