Change of Coordinate Systems

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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.
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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, <math> \theta </math>, 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 <math> \pi/2 </math> from the x-axis, giving it polar coordinates <math> (r,\theta) = (1,\pi/2) </math>. 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 <math> 2\pi </math> radians. Each of these circles is thus mapped to a straight line of length <math> 2\pi </math> 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.
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Many other such transformations exist and are useful throughout mathematics, such as mapping the points in a disk to a rectangle.
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|ImageDesc= Some of these mappings can be neatly represented in matrix notation, in the form
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<math> A\vec{x}=\vec{x'} </math>
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Where <math> \vec{x}</math> is the coordinate [[vector]] of our point in the original coordinate system and <math> \vec{x'} </math> is the coordinate vector of our point in the new coordinate system.
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For example the transformation in the basic description, doubling the value of the x-coordinate, is represented in this notation by
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: <math>\begin{bmatrix}
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2 & 0 \\
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0 & 1 \\
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\end{bmatrix}\vec{x} = \vec{x'}</math>
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As can be easily verified.
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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 <math> 2\pi </math>. 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.
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|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 14:40, 11 June 2009

Image:inprogress.png
Change of Coordinates
Field: Algebra
Created By: Brendan John
Website: [ ]

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

UNIQ1d654bbba8247 [...]

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.




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