Coordinate system

Ingeometry,acoordinate systemis a system that uses one or morenumbers,orcoordinates,to uniquely determine thepositionof thepointsor other geometric elements on amanifoldsuch asEuclidean space.^[1]^[2]The order of the coordinates is significant, and they are sometimes identified by their position in an orderedtupleand sometimes by a letter, as in "thex-coordinate ". The coordinates are taken to bereal numbersinelementary mathematics,but may becomplex numbersor 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]

Common coordinate systems

Number line

The simplest example of a coordinate system is the identification of points on alinewith real numbers using thenumber line.In this system, an arbitrary pointO(theorigin) is chosen on a given line. The coordinate of a pointPis defined as the signed distance fromOtoP,where the signed distance is the distance taken as positive or negative depending on which side of the linePlies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.^[4]

The number line

Cartesian coordinate system

TheCartesian coordinate systemin the plane

The Cartesian coordinate system in three-dimensional space

The prototypical example of a coordinate system is theCartesian coordinate system.In theplane,twoperpendicularlines are chosen and the coordinates of a point are taken to be the signed distances to the lines.^[5]In three dimensions, three mutuallyorthogonalplanes are chosen and the three coordinates of a point are the signed distances to each of the planes.^[6]This can be generalized to createncoordinates for any point inn-dimensional Euclidean space.

Depending on the direction and order of thecoordinate axes,the three-dimensional system may be aright-handedor a left-handed system.

Polar coordinate system

Another common coordinate system for the plane is thepolar coordinate system.^[7]A point is chosen as thepoleand 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 isrfor 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.

Cylindrical and spherical coordinate systems

Cylindrical coordinate system

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 therandθpolar 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]

Homogeneous coordinate system

A point in the plane may be represented inhomogeneous coordinatesby a triple (x,y,z) wherex/zandy/zare 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 planewithout 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.

Other commonly used systems

Some other common coordinate systems are the following:

Curvilinear coordinatesare a generalization of coordinate systems generally; the system is based on the intersection of curves.
- Orthogonal coordinates:coordinate surfacesmeet at right angles
- Skew coordinates:coordinate surfacesare not orthogonal
Thelog-polar coordinate systemrepresents 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 coordinatesare a way of representing lines in 3D Euclidean space using a six-tuple of numbers ashomogeneous coordinates.
Generalized coordinatesare used in theLagrangiantreatment of mechanics.
Canonical coordinatesare used in theHamiltoniantreatment of mechanics.
Barycentric coordinate systemas used forternary plotsand more generally in the analysis oftriangles.
Trilinear coordinatesare used in the context of triangles.

There are ways of describing curves without coordinates, usingintrinsic equationsthat use invariant quantities such ascurvatureandarc length.These include:

TheWhewell equationrelates arc length and thetangential angle.
TheCesàro equationrelates 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,circlesorspheres.For example,Plücker coordinatesare 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 coordinatesis 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]

Transformations

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 positivexaxis, then the coordinate transformation from polar to Cartesian coordinates is given byx=rcosθandy=rsinθ.

With everybijectionfrom 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 lines/curves

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 coordinatesare 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 lineused 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.

Many curves can occur as coordinate curves. For example, the coordinate curves ofparabolic coordinatesareparabolas.

Coordinate planes/surfaces

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 holdingρconstant in thespherical coordinate systemare 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 hypersurfacesare the(n− 1)-dimensional spaces resulting from fi xing a single coordinate of ann-dimensional coordinate system.^[14]

Coordinate maps

The concept of acoordinate map,orcoordinate chartis 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 ahomeomorphismfrom an open subset of a spaceXto 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 anatlascovering the space. A space equipped with such an atlas is called amanifoldand additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, adifferentiable manifoldis a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.

Orientation-based coordinates

Ingeometryandkinematics,coordinate systems are used to describe the (linear) position of points and theangular positionof 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 coordinatesof three points. These points are used to define the orientation of the axes of the local system; they are the tips of threeunit vectorsaligned 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 theHellenistic period,a variety of coordinate systems have been developed based on the types above, including:

Geographic coordinate system,thespherical coordinatesoflatitudeandlongitude
Projected coordinate systems,including thousands ofcartesian coordinate systems,each based on amap projectionto create a planar surface of the world or a region.
Geocentric coordinate system,a three-dimensionalcartesian coordinate systemthat models the earth as an object, and are most commonly used for modeling the orbits ofsatellites,including theGlobal Positioning Systemand othersatellite navigationsystems.

References

Citations

^Woods p. 1
^Weisstein, Eric W."Coordinate System".MathWorld.
^Weisstein, Eric W."Coordinates".MathWorld.
^Stewart, James B.;Redlin, Lothar; Watson, Saleem (2008).College Algebra(5th ed.).Brooks Cole.pp. 13–19.ISBN 978-0-495-56521-5.
^Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021).Calculus: Multivariable.John Wiley & Sons.p. 657.ISBN 978-1-119-77798-4.
^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).ISBN 978-0-387-18430-2.
^Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994).Calculus: Graphical, Numerical, Algebraic(Single Variable Version ed.). Addison-Wesley Publishing Co.ISBN 0-201-55478-X.
^Margenau, Henry;Murphy, George M. (1956).The Mathematics of Physics and Chemistry.New York City: D. van Nostrand. p.178.ISBN 978-0-88275-423-9.LCCN 55010911.OCLC 3017486.
^Morse, PM;Feshbach, H(1953).Methods of Theoretical Physics, Part I.New York: McGraw-Hill. p. 658.ISBN 0-07-043316-X.LCCN 52011515.
^Jones, Alfred Clement (1912).An Introduction to Algebraical Geometry.Clarendon.
^Hodge, W.V.D.;D. Pedoe(1994) [1947].Methods of Algebraic Geometry, Volume I (Book II).Cambridge University Press.ISBN 978-0-521-46900-5.
^Woods p. 2
^Tang, K. T. (2006).Mathematical Methods for Engineers and Scientists.Vol. 2. Springer. p. 13.ISBN 3-540-30268-9.
^Liseikin, Vladimir D. (2007).A Computational Differential Geometry Approach to Grid Generation.Springer. p. 38.ISBN 978-3-540-34235-9.
^Munkres, James R. (2000)Topology.Prentice Hall.ISBN 0-13-181629-2.
^Hanspeter Schaub;John L. Junkins(2003)."Rigid body kinematics".Analytical Mechanics of Space Systems.American Institute of Aeronautics and Astronautics. p. 71.ISBN 1-56347-563-4.

Sources

Voitsekhovskii, M.I.; Ivanov, A.B. (2001) [1994],"Coordinates",Encyclopedia of Mathematics,EMS Press
Woods, Frederick S. (1922).Higher Geometry.Ginn and Co. pp. 1ff.
Shigeyuki Morita; Teruko Nagase;Katsumi Nomizu(2001).Geometry of Differential Forms.AMS Bookstore. p. 12.ISBN 0-8218-1045-6.

External links

Hexagonal Coordinate Systems

[1] Woods p. 1

[2] Weisstein, Eric W."Coordinate System".MathWorld.

[3] Weisstein, Eric W."Coordinates".MathWorld.

[4] Stewart, James B.;Redlin, Lothar; Watson, Saleem (2008).College Algebra(5th ed.).Brooks Cole.pp. 13–19.ISBN 978-0-495-56521-5.

[5] Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021).Calculus: Multivariable.John Wiley & Sons.p. 657.ISBN 978-1-119-77798-4.

[6] 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).ISBN 978-0-387-18430-2.

[7] Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994).Calculus: Graphical, Numerical, Algebraic(Single Variable Version ed.). Addison-Wesley Publishing Co.ISBN 0-201-55478-X.

[8] Margenau, Henry;Murphy, George M. (1956).The Mathematics of Physics and Chemistry.New York City: D. van Nostrand. p.178.ISBN 978-0-88275-423-9.LCCN 55010911.OCLC 3017486.

[9] Morse, PM;Feshbach, H(1953).Methods of Theoretical Physics, Part I.New York: McGraw-Hill. p. 658.ISBN 0-07-043316-X.LCCN 52011515.

[10] Jones, Alfred Clement (1912).An Introduction to Algebraical Geometry.Clarendon.

[11] Hodge, W.V.D.;D. Pedoe(1994) [1947].Methods of Algebraic Geometry, Volume I (Book II).Cambridge University Press.ISBN 978-0-521-46900-5.

[12] Woods p. 2

[13] Tang, K. T. (2006).Mathematical Methods for Engineers and Scientists.Vol. 2. Springer. p. 13.ISBN 3-540-30268-9.

[14] Liseikin, Vladimir D. (2007).A Computational Differential Geometry Approach to Grid Generation.Springer. p. 38.ISBN 978-3-540-34235-9.

[15] Munkres, James R. (2000)Topology.Prentice Hall.ISBN 0-13-181629-2.

[16] Hanspeter Schaub;John L. Junkins(2003)."Rigid body kinematics".Analytical Mechanics of Space Systems.American Institute of Aeronautics and Astronautics. p. 71.ISBN 1-56347-563-4.

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