Coordinate system

The coordinate system is a complex of definitions that implements the coordinate method , that is, a way to determine the position and movement of a point or body using numbers or other symbols. The set of numbers that determine the position of a particular point is called the coordinates of this point.

In mathematics, coordinates are a collection of numbers associated with points of diversity in a certain map of a certain atlas .

In elementary geometry, coordinates are quantities that determine the position of a point on a plane and in space. On the plane, the position of the point is most often determined by the distances from two straight lines (coordinate axes) intersecting at one point (the origin) at a right angle; one of the coordinates is called the ordinate , and the other is called the abscissa . In space according to the Descartes system, the position of a point is determined by the distances from three coordinate planes intersecting at one point at right angles to each other, or by spherical coordinates , where the origin is at the center of the sphere.

In geography, coordinates are chosen as ( approximately ) a spherical coordinate system - latitude , longitude, and height above a known general level (for example, the ocean). See Geographic coordinates .

In astronomy, celestial coordinates are an ordered pair of angular quantities (for example, right ascension and declination ), with the help of which they determine the position of luminaries and auxiliary points on the celestial sphere. In astronomy, various celestial coordinate systems are used. Each of them essentially represents a spherical coordinate system (without a radial coordinate) with an appropriately selected fundamental plane and a reference point. Depending on the choice of the fundamental plane, the system of celestial coordinates is called horizontal ( horizon plane), equatorial ( equatorial plane), ecliptic ( ecliptic plane) or galactic (galactic plane).

The most used coordinate system is a rectangular coordinate system (also known as a Cartesian coordinate system ).

The coordinates on the plane and in space can be entered in an infinite number of different ways. Solving a particular mathematical or physical problem using the coordinate method, you can use various coordinate systems, choosing one of them in which the problem is solved easier or more convenient in this particular case. A well-known generalization of the coordinate system are reference frames and reference systems .

Content

Core Systems

This section explains the most commonly used coordinate systems in elementary mathematics.

Cartesian coordinates

The location of the point P on the plane is determined by the Cartesian coordinates using a pair of numbers $(x,y):$ ${\ displaystyle (x, y):}$

$x$ ${\ displaystyle x}$ - distance from point P to the y axis, taking into account the sign
$y$ ${\ displaystyle y}$ - distance from point P to the x axis, taking into account the sign

In space, you need 3 coordinates $(x,y,z):$ ${\ displaystyle (x, y, z):}$

$x$ ${\ displaystyle x}$ - distance from point P to the yz plane
$y$ ${\ displaystyle y}$ - distance from point P to the xz plane
$z$ ${\ displaystyle z}$ - distance from point P to the xy plane

Polar coordinates

Polar coordinates.

In the polar coordinate system used on the plane, the position of the point P is determined by its distance to the origin r = | OP | and the angle φ of its radius vector to the axis Ox .

In space, generalizations of polar coordinates are applied - cylindrical and spherical coordinate systems.

Cylindrical coordinates

Cylindrical coordinates.

Cylindrical coordinates - a three-dimensional analogue of the polar, in which the point P is an ordered triple $(r,\varphi ,z).$ ${\ displaystyle (r, \ varphi, z).}$ In terms of a Cartesian coordinate system,

$0\leqslant {r}$ ${\ displaystyle 0 \ leqslant {r}}$ ( radius ) is the distance from the z axis to the point P ,
$0\leqslant \varphi <360^{\circ }$ ${\ displaystyle 0 \ leqslant \ varphi <360 ^ {\ circ}}$ ( azimuth or longitude) - the angle between the positive ("plus") part of the x axis and the segment drawn from the pole to point P and projected onto the xy plane.
$z$ ${\ displaystyle z}$ (height) is equal to the Cartesian z- coordinate of the point P.

Note: in the literature the designation ρ is sometimes used for the first (radial) coordinate, the designation θ for the second (angular, or azimuthal), and the designation h for the third coordinate.

The polar coordinates have one drawback: the value of φ is not defined at r = 0 .

Cylindrical coordinates are useful for studying systems symmetrical about some axis. For example, a long cylinder with a radius R in Cartesian coordinates (with the z axis coinciding with the axis of the cylinder) has the equation $x^{2}+y^{2}=R^{2},$ ${\ displaystyle x ^ {2} + y ^ {2} = R ^ {2},}$ whereas in cylindrical coordinates it looks much simpler, like r = R.

Spherical coordinates

Spherical coordinates.

Spherical coordinates are a three-dimensional analogue of polar.

In a spherical coordinate system, the location of the point P is determined by three components: $(\rho ,\varphi ,\theta ).$ ${\ displaystyle (\ rho, \ varphi, \ theta).}$ In terms of a Cartesian coordinate system,

$0\leqslant \rho$ ${\ displaystyle 0 \ leqslant \ rho}$ (radius) is the distance from point P to the pole,
$0\leqslant \varphi \leqslant 360^{\circ }$ ${\ displaystyle 0 \ leqslant \ varphi \ leqslant 360 ^ {\ circ}}$ (azimuth or longitude) - the angle between the positive ("plus") axis x and the projection of the segment drawn from the pole to point P on the xy plane.
$0\leqslant \theta \leqslant 180^{\circ }$ ${\ displaystyle 0 \ leqslant \ theta \ leqslant 180 ^ {\ circ}}$ (latitude or polar angle) - the angle between the positive ("plus") semiaxis z and the segment drawn from the pole to point P.

Note: in the literature sometimes the azimuth is denoted by θ , and the polar angle is denoted by φ . Sometimes r is used instead of ρ for the radial coordinate. In addition, the range of angles for azimuth can be selected as (−180 °, + 180 °] instead of the range [0 °, + 360 °). Finally, the polar angle can be measured not from the positive direction of the z axis, but from the xy plane; in this case, it lies in the range [−90 °, + 90 °], and not in the range [0 °, 180 °]. Sometimes the order of coordinates in a triple is chosen different from that described; for example, polar and azimuthal angles can be rearranged.

The spherical coordinate system also has a drawback: φ and θ are not defined if ρ = 0; the angle φ is also not defined for the boundary values θ = 0 and θ = 180 ° (or for θ = ± 90 °, if the corresponding range for this angle is adopted).

To construct point P by its spherical coordinates, one needs to postpone a segment equal to ρ from the pole along the positive semiaxis z , turn it by an angle θ around the y axis in the direction of the positive x axis, and then turn by an angle θ around the z axis in the direction of the positive y axis.

Spherical coordinates are useful in studying systems that are symmetrical about a point. So, the equation of a sphere with radius R in Cartesian coordinates with the origin in the center of the sphere looks like $x^{2}+y^{2}+z^{2}=R^{2},$ ${\ displaystyle x ^ {2} + y ^ {2} + z ^ {2} = R ^ {2},}$ while in spherical coordinates it becomes much simpler: $\rho =R.$ ${\ displaystyle \ rho = R.}$

Other common coordinate systems

An affine (oblique) coordinate system is a rectilinear coordinate system in an affine space . On the plane, it is defined by the origin point O and two ordered non- collinear vectors , which represent an affine basis. Axes of coordinates in this case are called straight lines passing through the point of origin parallel to the basis vectors, which, in turn, determine the positive direction of the axes. In three-dimensional space , respectively, the affine coordinate system is defined by a triple of linearly independent vectors and a point of origin. To determine the coordinates of a point M, the coefficients of the expansion of the vector OM in the basis vectors are calculated ^[1] .
The barycentric coordinates were first introduced in 1827 by A. Mebius , who solved the problem of the center of gravity of the masses located at the vertices of the triangle . They are affine invariant; they are a special case of common homogeneous coordinates . A point with barycentric coordinates is located in an n- dimensional vector space E ⁿ , and the actual coordinates in this case refer to a fixed system of points that do not lie in an ( n −1) -dimensional subspace. Barycentric coordinates are also used in algebraic topology as applied to points of a simplex ^[2] .
Biangular coordinates - a special case of bicentric coordinates, a coordinate system on a plane defined by two fixed points C ₁ and C ₂ through which a line is drawn that acts as the abscissa axis. The position of a point P that does not lie on this line is determined by the angles PC ₁ C ₂ and PC ₂ C ₁ .
The bipolar coordinates ^{[3] are} characterized by the fact that in this case two families of circles with poles A and B act as coordinate lines on the plane, as well as a family of circles orthogonal to them. The transformation of bipolar coordinates into Cartesian rectangular is carried out using special formulas. Bipolar coordinates in space are called bipolar; in this case, the coordinate surfaces are spheres , surfaces formed by the rotation of the arcs of circles, as well as half-planes passing through the axis O _z ^[4] .
Bicentric coordinates are any coordinate system that is based on two fixed points and within which the position of some other point is determined, as a rule, by the degree of its removal or generally by the position relative to these two main points. Systems of this kind can be quite useful in certain areas of scientific research ^[5] ^[6] .
Bicylindrical coordinates - a coordinate system that is formed if the bipolar coordinate system on the O _{xy plane is} parallel transferred along the O _z axis. The coordinate surfaces in this case are the family of pairs of circular cylinders whose axes are parallel, the family of circular cylinders orthogonal to them, and also the plane. To translate bicylindrical coordinates into Cartesian coordinates for three-dimensional space, special formulas are also used ^[7] .
Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres, which are described by their radius , and two families of perpendicular cones located along the x and z axes ^[8] .
Rindler’s coordinates are used primarily in the framework of the theory of relativity and describe that part of flat space-time , which is usually called Minkowski space . In the special theory of relativity, a uniformly accelerating particle is in hyperbolic motion , and for each such particle in the Rindler coordinates, a reference point can be chosen relative to which it is at rest.
Parabolic coordinates are a two-dimensional orthogonal coordinate system in which coordinate lines are a set of confocal parabolas . A three-dimensional modification of parabolic coordinates is constructed by rotating a two-dimensional system around the axis of symmetry of these parabolas. Parabolic coordinates also have a certain range of potential practical applications: in particular, they can be used in relation to the Stark effect . Parabolic coordinates are connected by a certain relation with rectangular Cartesian ^[9] .
Projective coordinates exist, according to the name, in the projective space n _n ( K ) and represent a one-to-one correspondence between its elements and classes of finite subsets of elements of the body K characterized by equivalence and ordering properties. To determine the projective coordinates of projective subspaces, it suffices to determine the corresponding coordinates of the points of the projective space. In the general case, with respect to some basis, projective coordinates are introduced by purely projective means ^[10] .
A toroidal coordinate system is a three-dimensional orthogonal coordinate system obtained by rotating a two-dimensional bipolar coordinate system around an axis separating its two foci. The foci of the bipolar system, respectively, turn into a ring with radius a lying on the xy plane of the toroidal coordinate system, while the z axis becomes the axis of rotation of the system. The focal ring is also sometimes called the base circle ^[11] .
Trilinear coordinates are one of the patterns of homogeneous coordinates and are based on a given triangle, so the position of a point is determined relative to the sides of this triangle - mainly by the degree of distance from them, although other variations are possible. Trilinear coordinates can be relatively easily converted to barycentric; in addition, they are also convertible into two-dimensional rectangular coordinates, for which the corresponding formulas are used ^[12] .
Cylindrical parabolic coordinates - a three-dimensional orthogonal coordinate system, obtained as a result of the spatial transformation of a two-dimensional parabolic coordinate system. Coordinate surfaces, respectively, are confocal parabolic cylinders. Cylindrical parabolic coordinates are connected by a certain relation with rectangular ones; they can be applied in a number of fields of scientific research ^[13] .
Ellipsoidal coordinates are elliptical coordinates in space. The coordinate surfaces in this case are ellipsoids , single-sheeted hyperboloids , and also two-sheeted hyperboloids, the centers of which are located at the origin. The system is orthogonal. Eight points correspond to each triple of numbers, which are ellipsoidal coordinates, which are symmetrical to each other with respect to the planes of the system O _xyz ^[14] .

Transition from one coordinate system to another

Cartesian and polar

x=r\,\cos \varphi ,

{\ displaystyle x = r \, \ cos \ varphi,}

y=r\,\sin \varphi ,

{\ displaystyle y = r \, \ sin \ varphi,}

r={\sqrt {x^{2}+y^{2}}},

{\ displaystyle r = {\ sqrt {x ^ {2} + y ^ {2}}},}

\varphi =\operatorname {arctg} {\frac {y}{x}}+\pi u_{0}(-x)\,\operatorname {sgn} y,

{\ displaystyle \ varphi = \ operatorname {arctg} {\ frac {y} {x}} + \ pi u_ {0} (- x) \, \ operatorname {sgn} y,}

where u ₀ is the Heaviside function with $u_{0}(0)=0,$ ${\ displaystyle u_ {0} (0) = 0,}$ and sgn is the signum function . Here the functions u ₀ and sgn are used as “logical” switches, similar in meaning to the “if .. then” (if ... else) operators in programming languages. Some programming languages have a special function atan2 ( y , x ), which returns the correct φ in the necessary quadrant , defined by the x and y coordinates.

Cartesian and Cylindrical

x=r\,\cos \varphi ,

{\ displaystyle x = r \, \ cos \ varphi,}

y=r\,\sin \varphi ,

{\ displaystyle y = r \, \ sin \ varphi,}

z=z.\quad

{\ displaystyle z = z. \ quad}

r={\sqrt {x^{2}+y^{2}}},

{\ displaystyle r = {\ sqrt {x ^ {2} + y ^ {2}}},}

\varphi =\operatorname {arctg} {\frac {y}{x}}+\pi u_{0}(-x)\,\operatorname {sgn} y,

{\ displaystyle \ varphi = \ operatorname {arctg} {\ frac {y} {x}} + \ pi u_ {0} (- x) \, \ operatorname {sgn} y,}

z=z.\quad

{\ displaystyle z = z. \ quad}

{\begin{pmatrix}dx\\dy\\dz\end{pmatrix}}={\begin{pmatrix}r\cos \theta &-r\sin \varphi &0\\r\sin \theta &r\cos \varphi &0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}dr\\d\varphi \\dz\end{pmatrix}},

{\ displaystyle {\ begin {pmatrix} dx \\ dy \\ dz \ end {pmatrix}} = {\ begin {pmatrix} r \ cos \ theta & -r \ sin \ varphi & 0 \\ r \ sin \ theta & r \ cos \ varphi & 0 \\ 0 & 0 & 1 \ end {pmatrix}} \ cdot {\ begin {pmatrix} dr \\ d \ varphi \\ dz \ end {pmatrix}},}

{\begin{pmatrix}dr\\d\varphi \\dz\end{pmatrix}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{\sqrt {x^{2}+y^{2}}}}&{\frac {x}{\sqrt {x^{2}+y^{2}}}}&0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}dx\\dy\\dz\end{pmatrix}}.

{\ displaystyle {\ begin {pmatrix} dr \\ d \ varphi \\ dz \ end {pmatrix}} = {\ begin {pmatrix} {\ frac {x} {\ sqrt {x ^ {2} + y ^ { 2}}}} & {\ frac {y} {\ sqrt {x ^ {2} + y ^ {2}}}} & 0 \\ {\ frac {-y} {\ sqrt {x ^ {2} + y ^ {2}}}} & {\ frac {x} {\ sqrt {x ^ {2} + y ^ {2}}}} & 0 \\ 0 & 0 & 1 \ end {pmatrix}} \ cdot {\ begin {pmatrix } dx \\ dy \\ dz \ end {pmatrix}}.}

Cartesian and spherical

{x}=\rho \,\sin \theta \,\cos \varphi ,\quad

{\ displaystyle {x} = \ rho \, \ sin \ theta \, \ cos \ varphi, \ quad}

{y}=\rho \,\sin \theta \,\sin \varphi ,\quad

{\ displaystyle {y} = \ rho \, \ sin \ theta \, \ sin \ varphi, \ quad}

{z}=\rho \,\cos \theta ;\quad

{\ displaystyle {z} = \ rho \, \ cos \ theta; \ quad}

{\rho }={\sqrt {x^{2}+y^{2}+z^{2}}},

{\ displaystyle {\ rho} = {\ sqrt {x ^ {2} + y ^ {2} + z ^ {2}}},}

{\theta }=\arccos {\frac {z}{\rho }}=\operatorname {arctg} {\frac {\sqrt {x^{2}+y^{2}}}{z}},

{\ displaystyle {\ theta} = \ arccos {\ frac {z} {\ rho}} = \ operatorname {arctg} {\ frac {\ sqrt {x ^ {2} + y ^ {2}}} {z} },}

{\varphi }=\operatorname {arctg} {\frac {y}{x}}+\pi \,u_{0}(-x)\,\operatorname {sgn} y.

{\ displaystyle {\ varphi} = \ operatorname {arctg} {\ frac {y} {x}} + \ pi \, u_ {0} (- x) \, \ operatorname {sgn} y.}

{\begin{pmatrix}dx\\dy\\dz\end{pmatrix}}={\begin{pmatrix}\sin \theta \cos \varphi &\rho \cos \theta \cos \varphi &-\rho \sin \theta \sin \varphi \\\sin \theta \sin \varphi &\rho \cos \theta \sin \varphi &\rho \sin \theta \cos \varphi \\\cos \theta &-\rho \sin \theta &0\end{pmatrix}}\cdot {\begin{pmatrix}d\rho \\d\theta \\d\varphi \end{pmatrix}},

{\ displaystyle {\ begin {pmatrix} dx \\ dy \\ dz \ end {pmatrix}} = {\ begin {pmatrix} \ sin \ theta \ cos \ varphi & \ rho \ cos \ theta \ cos \ varphi & - \ rho \ sin \ theta \ sin \ varphi \\\ sin \ theta \ sin \ varphi & \ rho \ cos \ theta \ sin \ varphi & \ rho \ sin \ theta \ cos \ varphi \\\ cos \ theta & - \ rho \ sin \ theta & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} d \ rho \\ d \ theta \\ d \ varphi \ end {pmatrix}},}

{\begin{pmatrix}d\rho \\d\theta \\d\varphi \end{pmatrix}}={\begin{pmatrix}x/\rho &y/\rho &z/\rho \\{\frac {xz}{\rho ^{2}{\sqrt {x^{2}+y^{2}}}}}&{\frac {yz}{\rho ^{2}{\sqrt {x^{2}+y^{2}}}}}&{\frac {-(x^{2}+y^{2})}{\rho ^{2}{\sqrt {x^{2}+y^{2}}}}}\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\end{pmatrix}}\cdot {\begin{pmatrix}dx\\dy\\dz\end{pmatrix}}.

{\ displaystyle {\ begin {pmatrix} d \ rho \\ d \ theta \\ d \ varphi \ end {pmatrix}} = {\ begin {pmatrix} x / \ rho & y / \ rho & z / \ rho \\ { \ frac {xz} {\ rho ^ {2} {\ sqrt {x ^ {2} + y ^ {2}}}} & {\ frac {yz} {\ rho ^ {2} {\ sqrt {x ^ {2} + y ^ {2}}}}} & {\ frac {- (x ^ {2} + y ^ {2})} {\ rho ^ {2} {\ sqrt {x ^ {2} + y ^ {2}}}}} \\ {\ frac {-y} {x ^ {2} + y ^ {2}}} & {\ frac {x} {x ^ {2} + y ^ { 2}}} & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} dx \\ dy \\ dz \ end {pmatrix}}.}

Cylindrical and Spherical

{r}=\rho \,\sin \theta ,

{\ displaystyle {r} = \ rho \, \ sin \ theta,}

{\varphi }=\varphi ,\quad

{\ displaystyle {\ varphi} = \ varphi, \ quad}

{z}=\rho \,\cos \theta ;

{\ displaystyle {z} = \ rho \, \ cos \ theta;}

{\rho }={\sqrt {r^{2}+z^{2}}},

{\ displaystyle {\ rho} = {\ sqrt {r ^ {2} + z ^ {2}}},}

{\theta }=\operatorname {arctg} {\frac {z}{r}}+\pi \,u_{0}(-r)\,\operatorname {sgn} z,

{\ displaystyle {\ theta} = \ operatorname {arctg} {\ frac {z} {r}} + \ pi \, u_ {0} (- r) \, \ operatorname {sgn} z,}

{\varphi }=\varphi .\quad

{\ displaystyle {\ varphi} = \ varphi. \ quad}

{\begin{pmatrix}dr\\d\varphi \\dh\end{pmatrix}}={\begin{pmatrix}\sin \theta &\rho \cos \theta &0\\0&0&1\\\cos \theta &-\rho \sin \theta &0\end{pmatrix}}\cdot {\begin{pmatrix}d\rho \\d\theta \\d\varphi \end{pmatrix}},

{\ displaystyle {\ begin {pmatrix} dr \\ d \ varphi \\ dh \ end {pmatrix}} = {\ begin {pmatrix} \ sin \ theta & \ rho \ cos \ theta & 0 \\ 0 & 0 & 1 \\\ cos \ theta & - \ rho \ sin \ theta & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} d \ rho \\ d \ theta \\ d \ varphi \ end {pmatrix}},}

{\begin{pmatrix}d\rho \\d\theta \\d\varphi \end{pmatrix}}={\begin{pmatrix}{\frac {r}{\sqrt {r^{2}+z^{2}}}}&0&{\frac {z}{\sqrt {r^{2}+z^{2}}}}\\{\frac {-z}{r^{2}+z^{2}}}&0&{\frac {r}{r^{2}+z^{2}}}\\0&1&0\end{pmatrix}}\cdot {\begin{pmatrix}dr\\d\varphi \\dz\end{pmatrix}}.

{\ displaystyle {\ begin {pmatrix} d \ rho \\ d \ theta \\ d \ varphi \ end {pmatrix}} = {\ begin {pmatrix} {\ frac {r} {\ sqrt {r ^ {2} + z ^ {2}}}} & 0 & {\ frac {z} {\ sqrt {r ^ {2} + z ^ {2}}}} \\ {\ frac {-z} {r ^ {2} + z ^ {2}}} & 0 & {\ frac {r} {r ^ {2} + z ^ {2}}} \\ 0 & 1 & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} dr \\ d \ varphi \\ dz \ end {pmatrix}}.}

Geographic coordinate system

The geographic coordinate system provides the ability to identify any point on the surface of the globe by a combination of alphanumeric designations. As a rule, the coordinates are assigned in such a way that one of the pointers indicates the vertical position, and the other or a set of others - horizontally . The traditional set of geographical coordinates is latitude , longitude, and height ^[15] . The geographic coordinate system using the three listed pointers is orthogonal.

The latitude of a point on the Earth’s surface is defined as the angle between the plane of the equator and the straight line passing through this point in the form of a normal to the surface of the base ellipsoid, approximately coinciding in shape with the Earth. This line usually runs a few kilometers from the center of the Earth, with the exception of two cases: the poles and the equator (in these cases, it passes directly through the center). Lines connecting points of the same latitude are called parallels . 0 ° latitude corresponds to the equatorial plane, the North Pole of the Earth corresponds to 90 ° north latitude, the South - respectively, 90 ° south latitude. In turn, the longitude of a point on the Earth’s surface is defined as the angle in the east or west from the main meridian to another meridian passing through this point. Meridians connecting points of the same longitude are semi-ellipses converging at the poles. The meridian passing through the Royal Observatory in Greenwich , near London , is considered zero. As for the height, it is reckoned from the conditional surface of the geoid , which is an abstract spatial representation of the globe.

Notes

↑ Parkhomenko A.S. Affine coordinate system. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.
↑ Sklyarenko E. G. Barycentric coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.
↑ Weisstein, Eric W. Bipolar coordinates on the Wolfram MathWorld website.
↑ Dolgachev I.V., Pskovskikh V.A. Bipolar coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.
↑ R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.
↑ The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.
↑ Sokolov D. D. Bicylindrical coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.
↑ MathWorld description of conical coordinates
↑ MathWorld description of parabolic coordinates
↑ Wojciechowski M.I. Projective coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.
↑ MathWorld description of toroidal coordinates
↑ Weisstein, Eric W. Trilinear Coordinates on Wolfram MathWorld .
↑ MathWorld description of parabolic cylindrical coordinates
↑ Sokolov D. D. Ellipsoidal coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.
↑ A Guide to coordinate systems in Great Britain Archived on April 22, 2008. v 1.7 October 2007

Literature

Gelfand I.M., Glagoleva E.G., Kirillov A.A. Coordinate method. (inaccessible link) The fifth edition, stereotyped. Series: Library of the Physics and Mathematics School. Maths. Issue 1. M .: Nauka, 1973.
Delone N. B. Coordinates, in mathematics // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.

Links

Optional lesson in mathematics on the topic: "Different coordinate systems"

[1] Parkhomenko A.S. Affine coordinate system. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.

[2] Sklyarenko E. G. Barycentric coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.

[bip-3] Weisstein, Eric W. Bipolar coordinates on the Wolfram MathWorld website.

[4] Dolgachev I.V., Pskovskikh V.A. Bipolar coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.

[5] R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.

[6] The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.

[7] Sokolov D. D. Bicylindrical coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.

[8] MathWorld description of conical coordinates

[9] MathWorld description of parabolic coordinates

[10] Wojciechowski M.I. Projective coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.

[11] MathWorld description of toroidal coordinates

[12] Weisstein, Eric W. Trilinear Coordinates on Wolfram MathWorld .

[13] MathWorld description of parabolic cylindrical coordinates

[14] Sokolov D. D. Ellipsoidal coordinates. - Mathematical Encyclopedia. - M: Soviet Encyclopedia, 1977-1985.

[OSGB-15] A Guide to coordinate systems in Great Britain Archived on April 22, 2008. v 1.7 October 2007