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Encyclopedia > Coordinate system

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. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are put together to form an atlas covering the whole space. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Table of Geometry, from the 1728 Cyclopaedia. ... Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the Science of Nature. ... Engineering is the application of scientific and technical knowledge to solve human problems. ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ... A number is an abstract entity that represents a count or measurement. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... 2-dimensional renderings (ie. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... This article presents the essential definitions. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...


When the space has some additional algebraic structure, then the co-ordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...


Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. By convention the origin of the coordinate system in Cartesian coordinates is the point (0, 0, ..., 0), which may be assigned to any given point of Euclidean space. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


In physics, a scalar is a physical quantity which assumes a single value which is a "real" quantity independent of the coordinate system. In this sense coordinates are not scalars (although, of course, a scalar field can be defined which for one particular coordinate system corresponds to a particular coordinate). Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the Science of Nature. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... A physical quantity is either a quantity within physics that can be measured (e. ...


In some coordinate systems some points are associated with multiple tuples of coordinates, e.g. the origin in polar coordinates: r = 0 but θ can be any angle. This article describes some of the common coordinate systems that appear in elementary mathematics. ...

Contents


Examples

An example of a coordinate system is to describe a point P in the Euclidean space Rn by an n-tuple In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...

P = (r1, ..., rn)

of real numbers

r1, ..., rn.

These numbers r1, ..., rn are called the coordinates of the point P.


If a subset S of a Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S. That can be called a parametrization of the image, since it assigns numbers to points. That correspondence is unique only if the mapping is bijective. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...


The system of assigning longitude and latitude to geographical locations is a coordinate system. In this case the parametrization fails to be unique at the north and south poles. Longitude, sometimes denoted by the Greek letter λ, describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... Latitude, usually denoted symbolically by the Greek letter φ, gives the location of a place on Earth north or south of the Equator. ...


Transformations

A coordinate transformation is a conversion from one system to another, to describe the same space.


With every bijection from the space to itself two coordinate transformations can be associated: In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

  • 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, 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.


If the bijection is an involution, e.g. a reflection, then the two associated coordinate transformations are the same, e.g., in 1D, x becomes 7-x. In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ...


Examples of bijections include the invertible affine transformations. Of these, the similarity transformations preserve distance ratios, hence magnitude ratios, and angles, so that e.g. decomposition of a vector into perpendicular components is preserved. In the case that vector quantities are considered in relation to position and displacement, as in vector fields, a similarity transformation of space is normally accompanied by a corresponding linear transformation of the other vector quantities, to preserve angles between e.g. a force and a displacement, hence preserve e.g. dot products up to scaling. The transformation is linear because, as opposed to position, most vector quantities have a natural origin, e.g. zero force. However, velocity translation preserves the laws of motion, because an inertial frame of reference is preserved. (But if there is e.g. air-resistance, a velocity translation will affect tacitly assumed stationarity of air.) In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as... Several equivalence relations in mathematics are called similarity. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... An inertial frame is a coordinate system in which Newtons First Law of Motion is valid. ...


In diagrams showing vectors of multiple physical dimensions, e.g. forces and displacements, scaling of one kind of vectors does not affect relevant properties: a force and a displacement having the same length in a diagram has no particular significance.


Singularities

Please expand and improve this section. Further information might be found on this article's talk page or at Requests for expansion.

Some choices of coordinate systems may lead to paradoxes, for example, close to a black hole, but can be understood by changing the choice of coordinate system. At an actual mathematical singularity the coordinate system breaks down. A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...


Systems commonly used

Some coordinate systems are the following:

  • The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensionalHey everyone wuts up?? flat space, uses three numbers representing distances.
  • For any finite-dimensional vector space and any basis, the coefficients of the basis vectors can be used as coordinates. Changing the basis is a coordinate transformation, a linear transformation that can be summarized by a matrix, and is computationally the same as a mapping of points to other points keeping the bases the same: e.g. in 2D:
    • a clockwise rotation is a mapping of points to other points which changes the coordinates the same as keeping the points in place but rotating the coordinate axes anti-clockwise. The rotation of coordinate systems is covered in depth on wikibooks.
    • an expansion by a factor two in the direction of one basis vector is a mapping of points to other points which changes the coordinates the same as keeping the points in place but halving the magnitude of that basis vector (in both cases the corresponding coordinate is doubled).
    • a mapping of points to other points which distorts a rectangle to a parallelogram changes the coordinates the same as keeping the points in place but changing the basis vectors from being two sides of that parallelogram to perpendicular ones, two sides of that rectangle.
  • Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
  • The polar coordinate systems:
  • Generalized coordinates are used in the Lagrangian treatment of mechanics.
  • Canonical coordinates are used in the Hamiltonian treatment of mechanics.
  • Intrinsic coordinates describe a point upon a curve by the length of the curve to that point and the angle the tangent to that point makes with the x-axis.

Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... For the square matrix section, see square matrix. ... A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is part of the Wikimedia Foundation. ... A parallelogram. ... Perpendicular is a geometric term that may be used as a noun or adjective. ... Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ... It has been suggested that Polar graph be merged into this article or section. ... It has been suggested that Polar graph be merged into this article or section. ... The cylindrical coordinate system is a three-dimensional system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Map of Earth showing lines of latitude (horizontally) and longitude (vertically); large version (pdf) The geographic (earth-mapping) coordinate system expresses every horizontal position on Earth by two of the three coordinates of a spherical coordinate system which is aligned with the spin axis of the Earth. ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Important points in intrinsic coordinates Intrinsic coordinates is a coordinate system which defines points upon a curve partly by the nature of the tangents to the curve at that point. ...

Astronomical systems

In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky. ... The horizontal coordinate system is a celestial coordinate system that uses the observers local horizon as the fundamental plane. ... The equatorial coordinate system is probably the most widely used celestial coordinate system, whose equatorial coordinates are: declination () right ascension () -also RA-, or hour angle () -also HA- It is the most closely related to the geographic coordinate system, because they use the same fundamental plane, and the same poles. ... Earth (often referred to as The Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth in order of size. ... The ecliptic coordinate system is a celestial coordinate system that uses the ecliptic for its fundamental plane. ... The solar system comprises the Earths Sun and the retinue of celestial objects gravitationally bound to it. ... Many galaxies, including the Milky Way in which our Sun and Earth are located, are disk-shaped: the majority of their visible mass (excluding possible dark matter) lies very close to a plane. ... Note: This article contains special characters. ... Extragalactic astronomy is the branch of astronomy concerned with objects outside our own Milky Way Galaxy (the study of all astronomical objects which are not covered by galactic astronomy). ... Supergalactic coordinates are coordinates in a spherical coordinate system which was designed to have its equator aligned with the supergalactic plane, a major structure in the local universe formed by the preferential distribution of nearby galaxy clusters (such as the Virgo cluster, the Great Attractor and the Pisces-Perseus supercluster... Superclusters are large groupings of smaller galaxy groups and clusters, and are among the largest structures of the cosmos. ... NGC 4414, a typical spiral galaxy in the constellation Coma Berenices, is about 56,000 light years in diameter and approximately 60 million light years distant. ... The comoving distance or conformal distance of two objects in the universe is the distance divided by a time-varying scale factor representing the expansion of the universe. ... The particle horizon in cosmology is the distance from which particles (of positive mass or of zero mass) can have travelled to the observer in the age of the Universe. ...

See also


  Results from FactBites:
 
Cartesian coordinate system - Wikipedia, the free encyclopedia (1711 words)
In mathematics, the Cartesian coordinate system is used to uniquely determine each point in the plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.
In analytic geometry the Cartesian coordinate system is the foundation for the algebraic manipulation of geometrical shapes.
coordinate system - definition of coordinate system in Encyclopedia (678 words)
Some choices of coordinate systems may lead to paradoxes, for example, close to a fl hole, but can be understood by changing the choice of coordinate system.
Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height.
  More results at FactBites »

 
 

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