FACTOID # 10: The total number of state executions in 2005 was 60: 19 in Texas and 41 elsewhere. The racial split was 19 Black and 41 White.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
RELATED ARTICLES
People who viewed "Coordinates" also viewed:
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Coordinates

See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic.


In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. More generally, co-ordinates may sometimes be taken from rings or other ring-like algebraic structures.


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. Other coordinate systems do not, however, have a clear notion of origin. For example polar coordinates (r,θ) assign the point (x,y) = (0,0) the value r = 0 but θ any angle.

Contents

Examples

An example of a coordinate system is to describe a point P in the Euclidean space Rn by an n-tuple

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.


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.


Transformations

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


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.


Systems commonly used

Some coordinate systems are the following:

  • The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional 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 (http://en.wikibooks.org/wiki/Modern_Physics:Math:Vectors) 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 parallellogram 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

Astronomical systems

External links

  • Capětal city coordinates (http://times.clari.net.au/index.htm)

  Results from FactBites:
 
Polar Coordinates (408 words)
The 2-D polar coordinate system involves the distance from the origin and an azimuth angle.
The polar coordinates are useful in describing the human body motion since the essence of the human body motion is the joint motions.
In some cases, one may use the Cartesian coordinates in the sense of the polar coordinates, as shown on
Cuddly Coordinates: Baby Blankets and More (150 words)
Cuddly Coordinates specializes in one-of-a-kind blankets, quilts, room decor accessories, and clothing for the babies in your life.
Any item you request can have coordinating items made, also.
For example, a jacket to match a blanket or a frame to match a quilt.
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m