Change of Coordinate Systems

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As can be easily verified.
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.
+
The ellipse that is tilted relative to the coordinate axes is created by a combination of rotation and stretching, represented by the matrix
 +
: <math>\begin{bmatrix}
 +
 
 +
2cos(\theta) & -sin(\theta)\\
 +
2sin(\theta) & cos(\theta) \\
 +
\end{bmatrix}\vec{x} = \vec{x'}</math>
 +
 
 +
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 a different distance from the origin creates its own line in the polar system, and the collection of lines creates a rectangle.

Revision as of 13:48, 11 June 2009

Image:inprogress.png

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

UNIQ7057c4f149a6e [...]

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.

The ellipse that is tilted relative to the coordinate axes is created by a combination of rotation and stretching, represented by the matrix

\begin{bmatrix}

2cos(\theta) & -sin(\theta)\\
2sin(\theta) & cos(\theta)  \\
\end{bmatrix}\vec{x} = \vec{x'}

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 a 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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