# Change of Coordinate Systems

(Difference between revisions)
 Revision as of 12:30, 11 June 2009 (edit)← Previous diff Revision as of 13:40, 11 June 2009 (edit) (undo)Next diff → Line 7: Line 7: Under this transformation, a set of points would become stretched in the horizontal x-direction since each point becomes further from the vertical y-axis (except for points originally on the y-axis, which remain on the axis). A set of points that was originally contained in a circle in the old coordinates would be contained by a stretched-out ellipse in the new coordinate system, as shown in this page's main image. Under this transformation, a set of points would become stretched in the horizontal x-direction since each point becomes further from the vertical y-axis (except for points originally on the y-axis, which remain on the axis). A set of points that was originally contained in a circle in the old coordinates would be contained by a stretched-out ellipse in the new coordinate system, as shown in this page's main image. - Points can even be transferred to a different kind of coordinate system. A common example is mapping rectangular Cartesian Coordinates to '''Polar Coordinates'''. Each point's distance from the origin, R, and angle from the x-axis, $\theta$, are used as coordinates in the Polar Coordinate system. For example, the point (0,1) in the Cartesian system is a distance 1 from the origin and an angle $\pi/2$ from the x-axis, giving it polar coordinates $(r,\theta) = (1,\pi/2)$. Thus a disk in Cartesian Coordinates is mapped to a rectangle in Polar Coordinates: each origin-centered circle consists of points equidistant from the origin with angles from the x-axis ranging from zero to $2\pi$ radians. Each of these circles is thus mapped to a straight line of length $2\pi$ in Polar Coordinates. Since the distances from the origin of these circles range from zero to the radius of the disk, a set of lines is created in Polar Coordinates which together form a rectangle. + Many other such transformations exist and are useful throughout mathematics, such as mapping the points in a disk to a rectangle. + + |ImageDesc= Some of these mappings can be neatly represented in matrix notation, in the form + + $A\vec{x}=\vec{x'}$ + + Where $\vec{x}$ is the coordinate [[vector]] of our point in the original coordinate system and $\vec{x'}$ is the coordinate vector of our point in the new coordinate system. + + For example the transformation in the basic description, doubling the value of the x-coordinate, is represented in this notation by + : $\begin{bmatrix} + + 2 & 0 \\ + 0 & 1 \\ + \end{bmatrix}\vec{x} = \vec{x'}$ + + As can be easily verified. + + Often useful is mapping points from '''Cartesian Coordinates''' to '''[[Polar Coordinates]]'''. Such a mapping, as shown in this page's main image, can map a disk to a rectangle. Each origin-centered ring in the disk consists of points at constant distance from the origin and angles ranging from 0 to $2\pi$. These points create a vertical line in Polar Coordinates. Each ring at different distance from the origin creates its own line in the polar system, and the collection of lines creates a rectangle. + - |ImageDesc=Points in one space are undergo a transformation of some kind to be mapped to a points in another space. |AuthorName=Brendan John |AuthorName=Brendan John |Field=Algebra |Field=Algebra |InProgress=Yes |InProgress=Yes }} }}

## Revision as of 13:40, 11 June 2009

Change of Coordinates

The same object, here a circle, can look completely different depending on which coordinate system is used.

# Basic Description

It is a common practice in mathematics to use different coordinate systems to solve different problems. An example of a switch between coordinate systems follows: suppose we take a set of points in regular x-y Cartesian Coordinates, represented by ordered pairs such as (1,2), then multiply their x-components by two, meaning (1,2) in the old coordinates is matched with (2,2) in the new coordinates.

Under this transformation, a set of points would become stretched in the horizontal x-direction since each point becomes further from the vertical y-axis (except for points originally on the y-axis, which remain on the axis). A set of points that was originally contained in a circle in the old coordinates would be contained by a stretched-out ellipse in the new coordinate system, as shown in this page's main image.

Many other such transformations exist and are useful throughout mathematics, such as mapping the points in a disk to a rectangle.

# A More Mathematical Explanation

Some of these mappings can be neatly represented in matrix notation, in the form

UNIQ64bb03be74d3a [...]

Some of these mappings can be neatly represented in matrix notation, in the form

$A\vec{x}=\vec{x'}$

Where $\vec{x}$ is the coordinate vector of our point in the original coordinate system and $\vec{x'}$ is the coordinate vector of our point in the new coordinate system.

For example the transformation in the basic description, doubling the value of the x-coordinate, is represented in this notation by

$\begin{bmatrix} 2 & 0 \\ 0 & 1 \\ \end{bmatrix}\vec{x} = \vec{x'}$

As can be easily verified.

Often useful is mapping points from Cartesian Coordinates to Polar Coordinates. Such a mapping, as shown in this page's main image, can map a disk to a rectangle. Each origin-centered ring in the disk consists of points at constant distance from the origin and angles ranging from 0 to $2\pi$. These points create a vertical line in Polar Coordinates. Each ring at different distance from the origin creates its own line in the polar system, and the collection of lines creates a rectangle.