Change of Coordinate Systems

Change of Coordinates
Field: Algebra
Image Created By: Brendan John

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

'"`UNIQ--math-000 [...]

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

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

Some very useful mappings cannot be represented in matrix form, such as 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 these lines creates a rectangle.

The transformation from Cartesian coordinates to Polar Coordinate can be represented algebraically by

$\begin{bmatrix} r\\ \theta\\ \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2}\\ \arctan{y/x}\\ \end{bmatrix}$
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