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Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level is a coordinate which is useful for describing points near the surface of the earth. A coordinate system, in a plane or in space, is a systematic method of assigning a pair or a triple of numbers to each point in the plane or in space (respectively) which describe its position uniquely. For example, the triple consisting of latitude, longitude and altitude (height above sea level) define a coordinate system near to the surface of the earth. Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ...
Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ...
Altitude is the elevation of an object from a known level or datum. ...
 The Cartesian coordinate system. Coordinates may be defined in more general contexts. For example, if one is not interested in height, then latitude and longitude form a coordinate system on the surface of the earth, which is (approximately) a sphere. Coordinates such as these are also important in astronomy for describing the location of objects in the (night) sky: see Celestial coordinate systems for further examples. For simplicity, however, this article will restrict attention to coordinate systems in a plane and in space. Image File history File links Cartesian-coordinate-system. ...
Image File history File links Cartesian-coordinate-system. ...
A sphere is a perfectly symmetrical geometrical object. ...
A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy (also frequently referred to as astrophysics) is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as the cosmic background radiation). ...
In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky. ...
Cartesian coordinates
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In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represented by a pair of numbers (x,y). Fig. ...
- x is the signed distance from the y-axis to the point P, and
- y is the signed distance from the x-axis to the point P.
In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represented by a triple of numbers (x,y,z). - x is the signed distance from the yz-plane to the point P,
- y is the signed distance from the xz-plane to the point P, and
- z is the signed distance from the xy-plane to the point P.
 Image File history File links This is a lossless scalable vector image. ...
Polar coordinates The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of curvilinear coordinates. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...
In mathematics the term subtended usually refers to the direct relationship between an angle and its arc length. ...
An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...
Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ...
The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional).
Circular coordinates -
The circular coordinate system, commonly referred to as the polar coordinate system, is a two-dimensional polar coordinate system, defined by an origin, O, and a ray (or semi-infinite line) L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis). A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...
A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...
A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ...
Fig. ...
 In this picture x is L and the Cartesian coordinate axis is used for the purpose of illustration In the circular coordinate system, a point P is represented by a pair (r, θ). Using terms of the Cartesian coordinate system, Image File history File links This is a lossless scalable vector image. ...
Image File history File links This is a lossless scalable vector image. ...
Fig. ...
 (radius) is the distance from the origin to the point P, and (azimuth) is the angle between the positive x-axis and the line from the origin to the point P. Possible coordinate transformations from one circular coordinate system to another include: Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, from the north toward the east (i. ...
- change of zero direction (such as making north the zero direction)
- changing from the angle increasing counterclockwise to increasing clockwise or conversely (as in a compass)
- change of scale
and combinations. More generally, transformations of the corresponding Cartesian coordinates can be translated into transformations from one circular coordinate system to another by basically transforming to Cartesian coordinates, transforming those, and transforming back to circular coordinates. This is e.g needed for: - change of origin
- change of scale in one direction
A minor change is changing the range to e.g. 
Circular coordinates can be convenient in situations where only the distance, or only the direction to a fixed point matters, rotations about a point, etc. (by taking the special point as the origin).
A complex number can be viewed as a point or a position vector on a plane, the so-called complex plane or Argand diagram. Here the circular coordinates are r = |z|, called the absolute value or modulus of z, and φ = arg(z), called the complex argument of z. These coordinates (mod-arg form) are especially convenient for complex multiplication and powers. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
The Modulus-Argument form (Mod-Arg) of a complex number in mathematics is given by the modulus times the number e raised to the power of i times the argument. ...
Cylindrical coordinates -
The cylindrical coordinate system is a three-dimensional polar coordinate system. 2 points plotted with cylindrical coordinates The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ...
 Image File history File links This is a lossless scalable vector image. ...
In the cylindrical coordinate system, a point P is represented by a triple (r, θ, h). Using terms of the Cartesian coordinate system, Fig. ...
- (radius) is the distance between the z-axis and the point P,
- (azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane, and
- h (height) is the signed distance from xy-plane to the point P.
- Note: some sources use z for h; there is no "right" or "wrong" convention, but it is necessary to be aware of the convention being used.
Cylindrical coordinates involve some redundancy; θ loses its significance if r = 0. Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, from the north toward the east (i. ...
Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ...
Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis. For example the infinitely long cylinder that has the Cartesian equation x2 + y2 = c2 has the very simple equation r = c in cylindrical coordinates.
Spherical coordinates The spherical coordinate system is a three-dimensional polar coordinate system. A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle...
 Image File history File links This is a lossless scalable vector image. ...
In the spherical coordinate system, a point P is represented by a triple (ρ,θ,φ). Using terms of the Cartesian coordinate system, Fig. ...
 (radius) is the distance between the point P and the origin, (zenith, colatitude or polar angle) is the angle between the z-axis and the line from the origin to the point P, and- (azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane.
There are different conventions for the exact letters used for the angles. Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
In broad terms, the zenith is the direction pointing directly above a particular location (perpendicular, orthogonal). ...
Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ...
A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...
Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, from the north toward the east (i. ...
Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ...
The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates. Stereographic projection of the hyperspheres parallels (red), meridians (blue) and hypermeridians (green). ...
Transformations between coordinate systems - main article: List of canonical coordinate transformations
Because there are many different possible coordinate systems for describing points in the plane or in space, it is important to understand how they are related. Such relations are described by coordinate transformations which give formulae 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. This is a list of canonical coordinate transformations. ...
See also In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ...
In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...
Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ...
This is a list of some vector calculus formulae of general use in working with standard coordinate systems. ...
Parabolic coordinates are an alternative system of coordinates for three dimensions. ...
A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...
// Vector fields in cylindrical coordinates Vectors are defined in cylindrical coordinates by (Ï,φ,z), where Ï is the length of the vector projected onto the X-Y-plane, φ is the angle of the projected vector with the positive X-axis (0 ≤ φ < 2Ï€), z is the regular z-coordinate. ...
Spherical coordinates In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky. ...
Euler angles are a means of representing the spatial orientation of an object. ...
In gyroscopic devices controlled by Euler mechanics or Euler angles, gimbal lock is caused by the alignment of two of the three gimbals together so that one of the rotation references (pitch/yaw/roll, often yaw) is cancelled. ...
Spherical Harmonic is a fantasy novel by Catherine Asaro which tells the story of Pharaoh Dyhianna (Dehya) Selei, ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord. ...
Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ...
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