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This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space. A coordinate system is a plane or space where the origin and axes are defined so that coordinates can be measured. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...
Cartesian coordinates
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In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components (x,y).
- 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 represent by a tuple of three components (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.
For advanced topics, please refer to Cartesian coordinate system. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
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. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
This article is about angles in geometry. ...
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, often referred to simply as the polar coordinate system, is a two-dimensional polar coordinate system, defined by an origin, O, and a 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). Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
 Image File history File links Example for circular coordinate system File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
In the circular coordinate system, a point P is represented by a tuple of two components (r,θ). Using terms of the Cartesian coordinate system, Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
- (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.
RADIUS (Remote Authentication Dial In User Service) is an Authentication, Authorization and Accounting (AAA) 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 point, toward the East, i. ...
Cylindrical coordinates The cylindrical coordinate system is a three-dimensional polar coordinate system.
 Image File history File links Example for a cylindrical coordinate system File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
In the cylindrical coordinate system, a point P is represented by a tuple of three components (r,θ,h). Using terms of the Cartesian coordinate system, Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
- (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. RADIUS (Remote Authentication Dial In User Service) is an Authentication, Authorization and Accounting (AAA) 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 point, toward the East, i. ...
Map of Earth showing curved lines of longitude Longitude, sometimes denoted λ, 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.
 Image File history File links Download high resolution version (1000x1000, 16 KB)Spherical Coordinates Drawn by the Jacob Rus in Adobe Illustrator using fonts from AMS LaTeX, to replace earlier drawing by Cryogen File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to...
In the spherical coordinate system, a point P is represented by a tuple of three components (ρ,φ,θ). Using terms of the Cartesian coordinate system, Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
- (radius) is the distance between the point P and the origin,
- (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.
NB: The above convention is the standard used by American mathematicians and American calculus textbooks. However, most physicists, engineers, and non-American mathematicians interchange the symbols φ and θ above, using φ to denote the azimuth and θ the colatitude. One should be very careful to note which convention is being used by a particular author. It should be noted that, regardless of how one labels the coordinates, one argument against the conventional American mathematical definition is the fact that it produces a left-handed coordinate system, rather than the usual convention of a right-handed coordinate system. RADIUS (Remote Authentication Dial In User Service) is an Authentication, Authorization and Accounting (AAA) protocol for applications such as network access or IP mobility. ...
Latitude, denoted by the Greek letter φ, gives the location of a place on Earth north or south of the Equator. ...
Azimuth is the horizontal component of a direction (compass direction), measured around the horizon from the North point, toward the East, i. ...
Map of Earth showing curved lines of longitude Longitude, sometimes denoted λ, describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ...
The spherical coordinate system also involves some redundancy; φ loses its significance if ρ = 0, and θ loses its significance if ρ = 0 or φ = 0 or .
To construct a point from its spherical coordinates: from the origin, go ρ along the positive z-axis, rotate φ about y-axis toward the direction of the positive x-axis, and rotate θ about the z-axis toward the direction of the positive y-axis.
Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the very simple equation ρ = c in spherical coordinates.
Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics. Another application is ergonomic design, where ρ is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...
The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates. A hypersphere is a higher-dimensional analogue of a sphere. ...
Conversion between coordinate systems
Cartesian and circular where u0 is the Heaviside step function with u0(0) = 0 and sgn is the signum function. Here the u0 and sgn functions are being used as "logical" switches which are used as shorthand substitutes for several if ... then statements. Some computer languages include a bivariate arctangent function atan2(y,x) which finds the value for θ in the correct quadrant given x and y. The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of control theory and signal processing to represent a signal...
Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...
Electrical switches. ...
Cartesian and cylindrical
Cartesian and spherical
Cylindrical and spherical
See also Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ...
See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system, such that the transformation is locally invertible at each point. ...
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. ...
In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but this...
Vector fields in cylindrical coordinates Vectors are defined in cylindrical coordinates by (ρ,φ,z), where ρ 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 < 2π), z is the regular z...
Spherical coordinates In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky. ...
Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ...
Gimbal lock, in gyroscopic devices controlled by Euler mechanics or Euler angles, 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. ...
In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...
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. ...
External links - Frank Wattenberg has made some nice animations illustrating spherical and cylindrical coordinate systems.
- http://www.physics.oregonstate.edu/bridge/papers/spherical.pdf is a description of the different conventions in use for naming components of spherical coordinates, along with a proposal for standardizing this.
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