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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 numbers ...
s, or coordinates, to uniquely determine the
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
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 * Point ...
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 ...
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 Euclidean sp ...
. 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 for ...
s or elements of a more abstract system such as a commutative ring. 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.


Common coordinate systems


Number line

The simplest example of a coordinate system is the identification of points on a line 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 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 can ...
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 are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates. * Generalized coordinates are used in the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
treatment of mechanics. * Canonical coordinates are used in the Hamiltonian treatment of mechanics. * Barycentric coordinate system as used for ternary plots 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- colli ...
s. * Trilinear coordinates are used in the context of triangles. There are ways of describing curves without coordinates, using intrinsic equations 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 relates arc length and the tangential angle. * 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 const ...
s or
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s. For example, Plücker coordinates 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 In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point. Lines in the plane There are several possible ways to specify the position of ...
'' 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 Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Dual ...
''.


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 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 lines, they may be called coordinate lines. In Cartesian coordinate systems, coordinates lines are mutually orthogonal, and are known as '' coordinate axes''. 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''. 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'' me ...
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 isomor ...
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 geograp ...
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 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, 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 imagin ...
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 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 vectors 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 vari ...
, 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 po ...
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 ...
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 longit ...
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 ...
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 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 hi ...
systems.


See also

* Absolute angular momentum * Alphanumeric grid * Axes conventions in engineering * Celestial coordinate system * Coordinate-free * Fractional coordinates * Frame of reference *
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 rotati ...
* Grid reference * Nomogram, graphical representations of different coordinate systems * Reference system * Rotation of axes * Translation of axes


Relativistic coordinate systems

* Eddington–Finkelstein coordinates * Gaussian polar coordinates * Gullstrand–Painlevé coordinates * Isotropic coordinates * Kruskal–Szekeres coordinates * Schwarzschild coordinates


References


Citations


Sources

* * *


External links


Hexagonal Coordinate Systems
{{Authority control Analytic geometry