Change of Coordinate Systems

From Math Images

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| -                                      | {{Image Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                       | {{Image Description Ready                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | |ImageName=Change of Coordinates                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                         | |ImageName=Change of Coordinates                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | |Image=Coordchange.JPG                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                         | |Image=Coordchange.JPG                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | |ImageIntro=The same object, here a circle, can look completely different depending on which coordinate system is used.                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                       | |ImageIntro=The same object, here a disk, can look completely different depending on which coordinate system is used.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        | |ImageDescElem=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.                                                                                                        |                                                         | |ImageDescElem=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.                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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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.                                                                                                            | +                                                       | This transformation is shown below in the following to images. In the image on the left we have a square with four points marked: (0.2,0.7), (0.3,0.9), (0.4,0.3), and (0.9,0.3). The image on the right has undergone the transformation: instead of having a square, we now have a rectangle with the points (0.4,0.7), (0.6,0.9), (0.8,0.3), and (1.8,0.3). We can see that in call cases, y dimensions and coordinates remain the same, while all x coordinates and dimensions are doubled.                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | [[Image:Unstretched.png|left|220px]] [[Image:Stretched.png|400px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
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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.                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                       | <br><br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      | |ImageDesc= Some of these mappings can be neatly represented in matrix notation, in the form                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                       | Under this transformation, a set of points would be stretched out 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).                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | We can also see that 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 the top two figures of 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.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | |ImageDesc=Some of these mappings can be neatly represented by [[Vector|vectors]] and [[Matrix|matrices]], in the form                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        | <math> A\vec{x}=\vec{x'} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                         | <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.                                                                                                                                                                                                                                                                                                                                       | +                                                       | Where <math> \vec{x}</math> is the <balloon title="A vector for which each coordinate represents a component.">coordinate vector</balloon>  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                                                                                                                                                                                                                                                                                                                                                                                                           |                                                         | 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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|                                        | As can be easily verified.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                         | As can be easily verified.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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| -                                      | The ellipse that is tilted relative to the coordinate axes is created by a combination of rotation and stretching, represented by the matrix                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                       | In the main image of the page, 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}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                         | : <math>\begin{bmatrix}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        | \end{bmatrix}\vec{x} = \vec{x'}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                         | \end{bmatrix}\vec{x} = \vec{x'}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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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 a different distance from the origin creates its own line in the polar system, and the collection of lines creates a rectangle. | +                                                       | 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.  We can think of the disk as a series of rings wrapped around the origin, and the rectangle as a series of lines. Each of these rings is a different distance from the origin, and gets mapped to a different line within the 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 these lines creates a rectangle.                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | The transformation from Cartesian coordinates to Polar Coordinate can be represented algebraically by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \begin{bmatrix}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | r\\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \theta\\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \end{bmatrix}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | =                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \begin{bmatrix}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \sqrt{x^2 + y^2}\\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \arctan{y/x}\\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | \end{bmatrix}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | ===Three-Dimensional Coordinates===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | In 3 dimensions, similar coordinate systems and transformations between them exist.  Three common systems are rectangular, cylindrical and spherical coordinates:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :*Rectangular Coordinates use standard <math> x,y,z </math> coordinates, where the three coordinates represent left-right position, up-down position, and forward-backward position, respectively. These three directions are mutually <balloon title="At right angles to each other">perpendicular </balloon>.                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | [[Image:Cylindrical.png|200px|right]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :*Cylindrical Coordinates use <math> r,\theta,z</math>, where <math> r, \theta </math> are the same as two-dimensional polar coordinates and ''z'' is distance from the x-y plane as shown on the right.                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <br><br><br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | [[Image:Spherical.png|200px|left]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <br><br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :*Spherical Coordinates use <math> \rho, \theta, \phi </math>, where <math> \rho </math> is the distance from the origin, <math> \theta </math> is rotation from the positive x-axis as in polar coordinates, <math> \phi </math> and is rotation from the positive z-axis. Note that this standard varies from discipline to discipline. For example, the standard in physics is to switch the <math> \phi </math> and <math> \theta </math> labeling. Always be aware of what standard you should be using given a particular textbook or course. The mathematics standard noted above and shown in the image on the left is used for this page. |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <br><br>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | ====Converting between these coordinates====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | The conversion from rectangular (Cartesian) coordinates to cylindrical coordinates is almost identical to the conversion between Crtesian coordinates and polar coordinates.                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> r= \sqrt{x^2+y^2} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \theta=\tan^{-1}\left(y/x\right) </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> z=z </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | {{Switch|link1=Click to show example|link2=Click to hide example |1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | We can calculate the cylindrical coordinates for the point given by (1,2,3) in Cartesian coordinates.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> r=\sqrt{x^2+y^2}=\sqrt{1^2+2^2}=\sqrt{5} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \theta =\tan^{-1}\left(y/x\right)=\tan^{-1}\left(2/1\right)\approx 1.1 </math> radians                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> z=3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | So we have the point <math> (\sqrt{5}, 1.1,3 )</math> in cylindrical coordinates.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | The conversion from cylindrical coordinates to Cartesian coordinates is given by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> x=r \cos \theta </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> y=r \sin \theta </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> z=z </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | {{Switch|link1=Click to show example|link2=Click to hide example |1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | Now we can convert the point <math> (\sqrt{5}, 1.1,3 )</math> in cylindrical coordinates back to Cartesian coordinates.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> x=\sqrt{5} \cos (1.1)=\sqrt{5}(0.454)\approx 1 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> y= \sqrt{5} \sin (1.1)=\sqrt{5}(0.891)\approx 2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> z=3 </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | We see that we do indeed get back the point (1,2,3). The approximately equal to signs are due to rounding involved in dealing with the square root of five and sine and cosine. }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | In order to go from Cartesian to spherical coordinates, we have                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \rho=\sqrt{x^2+y^2+z^2} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \tan\phi=\frac{\sqrt{x^2+y^2}}{z} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \tan \theta = y/x </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | {{Switch|link1=Click to show example|link2=Click to hide example |1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | Starting out with the point (2,1,2) in Cartesian coordinates, we find that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \rho=\sqrt{x^2+y^2+z^2}=\sqrt{2^2+1^2+2^2}=\sqrt{4+1+4}=\sqrt{9}=3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \tan\phi=\frac{\sqrt{x^2+y^2}}{z}=\frac{\sqrt{2^1+1^2}}{2}=\frac{\sqrt{5}}{2} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | ::<math> \rightarrow \phi=\tan^{-1}\left(\sqrt{5}/2\right)=0.841</math> radians                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | : <math> \tan \theta = y/x=1/2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | ::<math> \rightarrow \theta=\tan^{-1}=0.464 </math> radians                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | So we have the point (3,0.841,0.464) in spherical coordinates}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | The transformation from spherical coordinates to Cartesian coordinates is given by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math>x=\rho \sin \phi \cos \theta </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> y=\rho \sin \phi \sin \theta </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> z= \rho \cos \phi </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | {{Switch|link1=Click to show example|link2=Click to hide example |1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | We can convert our previous result back to Cartesian coordinates:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math>x=\rho \sin \phi \cos \theta=3\sin(0.841)\cos(0.464)\approx 2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> y=\rho \sin \phi \sin \theta =3\sin(0.841)\sin(0.464)\approx 1 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> z= \rho \cos \phi =3 cos(0.841)\approx 2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | Again, we retrieve our original ordered triple (2,1,2) by rounding.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | We can also write the transformation from cylindrical coordinates to spherical coordinates:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \rho=\sqrt{r^2+z^2} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \tan \phi = z/r </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \theta = \theta </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | {{Switch|link1=Click to show example|link2=Click to hide example |1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | Beginning with the point (3,\pi/3,4) in cylindrical coordinates, we see that                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \rho=\sqrt{r^2+z^2}=\sqrt{3^2+4^2}=\sqrt{25}=5 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \tan \phi = z/r=4/3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | ::<math> \rightarrow \phi=\tan^{-1}(.75)=0.644 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \theta = \pi/3 </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | So the point in spherical coordinates is (5,\pi/3,0.644). }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | Finally, the transformation from spherical to cylindrical coordinates is given by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> r=\rho \sin \phi </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \theta =\theta </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> z=\rho \cos \phi </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | {{Switch|link1=Click to show example|link2=Click to hide example |1= |2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | For one last example, we will take our previous result and transform it back to cylindrical coordinates                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> r=\rho \sin \theta=5 \sin (0.644)\approx 3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> \theta =\pi/3</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | :<math> z=\rho \cos \theta5 \cos (0.644)\approx 4 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | Therefore the point in cylindrical coordinates is (3, \pi/3, 4) as expected. }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | ===Interactive Demonstration===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | <htmlet>ChangeOfCoordsApp</htmlet>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | *add examples of transformations between three dimensional coordinate systems.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        | |AuthorName=Brendan John                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                         | |AuthorName=Brendan John                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| -                                      | |Field=Algebra                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                       | |Field=Calculus                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      | |InProgress=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                       | |InProgress=No                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                         | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |

Current revision

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

Change of Coordinates

The same object, here a disk, 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.

This transformation is shown below in the following to images. In the image on the left we have a square with four points marked: (0.2,0.7), (0.3,0.9), (0.4,0.3), and (0.9,0.3). The image on the right has undergone the transformation: instead of having a square, we now have a rectangle with the points (0.4,0.7), (0.6,0.9), (0.8,0.3), and (1.8,0.3). We can see that in call cases, y dimensions and coordinates remain the same, while all x coordinates and dimensions are doubled.

Under this transformation, a set of points would be stretched out 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).

We can also see that 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 the top two figures of 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

[Click to view A More Mathematical Explanation]

Some of these mappings can be neatly represented by vectors and matrices, in th [...]

[Click to hide A More Mathematical Explanation]

Some of these mappings can be neatly represented by vectors and matrices, 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.

In the main image of the page, 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. We can think of the disk as a series of rings wrapped around the origin, and the rectangle as a series of lines. Each of these rings is a different distance from the origin, and gets mapped to a different line within the 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

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

Three-Dimensional Coordinates

In 3 dimensions, similar coordinate systems and transformations between them exist. Three common systems are rectangular, cylindrical and spherical coordinates:

• Rectangular Coordinates use standard $x,y,z$ coordinates, where the three coordinates represent left-right position, up-down position, and forward-backward position, respectively. These three directions are mutually perpendicular .

• Cylindrical Coordinates use $r,\theta,z$, where $r, \theta$ are the same as two-dimensional polar coordinates and z is distance from the x-y plane as shown on the right.

• Spherical Coordinates use $\rho, \theta, \phi$, where $\rho$ is the distance from the origin, $\theta$ is rotation from the positive x-axis as in polar coordinates, $\phi$ and is rotation from the positive z-axis. Note that this standard varies from discipline to discipline. For example, the standard in physics is to switch the $\phi$ and $\theta$ labeling. Always be aware of what standard you should be using given a particular textbook or course. The mathematics standard noted above and shown in the image on the left is used for this page.

Converting between these coordinates

The conversion from rectangular (Cartesian) coordinates to cylindrical coordinates is almost identical to the conversion between Crtesian coordinates and polar coordinates.

$r= \sqrt{x^2+y^2}$
$\theta=\tan^{-1}\left(y/x\right)$
$z=z$
[Click to show example]
[Click to hide example]
We can calculate the cylindrical coordinates for the point given by (1,2,3) in Cartesian coordinates.
$r=\sqrt{x^2+y^2}=\sqrt{1^2+2^2}=\sqrt{5}$
$\theta =\tan^{-1}\left(y/x\right)=\tan^{-1}\left(2/1\right)\approx 1.1$ radians
$z=3$
So we have the point $(\sqrt{5}, 1.1,3 )$ in cylindrical coordinates.

The conversion from cylindrical coordinates to Cartesian coordinates is given by

$x=r \cos \theta$
$y=r \sin \theta$
$z=z$.
[Click to show example]
[Click to hide example]
Now we can convert the point $(\sqrt{5}, 1.1,3 )$ in cylindrical coordinates back to Cartesian coordinates.
$x=\sqrt{5} \cos (1.1)=\sqrt{5}(0.454)\approx 1$
$y= \sqrt{5} \sin (1.1)=\sqrt{5}(0.891)\approx 2$
$z=3$.
We see that we do indeed get back the point (1,2,3). The approximately equal to signs are due to rounding involved in dealing with the square root of five and sine and cosine.

In order to go from Cartesian to spherical coordinates, we have

$\rho=\sqrt{x^2+y^2+z^2}$
$\tan\phi=\frac{\sqrt{x^2+y^2}}{z}$
$\tan \theta = y/x$
[Click to show example]
[Click to hide example]
Starting out with the point (2,1,2) in Cartesian coordinates, we find that
$\rho=\sqrt{x^2+y^2+z^2}=\sqrt{2^2+1^2+2^2}=\sqrt{4+1+4}=\sqrt{9}=3$
$\tan\phi=\frac{\sqrt{x^2+y^2}}{z}=\frac{\sqrt{2^1+1^2}}{2}=\frac{\sqrt{5}}{2}$
$\rightarrow \phi=\tan^{-1}\left(\sqrt{5}/2\right)=0.841$ radians
$\tan \theta = y/x=1/2$
$\rightarrow \theta=\tan^{-1}=0.464$ radians
So we have the point (3,0.841,0.464) in spherical coordinates

The transformation from spherical coordinates to Cartesian coordinates is given by

$x=\rho \sin \phi \cos \theta$
$y=\rho \sin \phi \sin \theta$
$z= \rho \cos \phi$.
[Click to show example]
[Click to hide example]
We can convert our previous result back to Cartesian coordinates:
$x=\rho \sin \phi \cos \theta=3\sin(0.841)\cos(0.464)\approx 2$
$y=\rho \sin \phi \sin \theta =3\sin(0.841)\sin(0.464)\approx 1$
$z= \rho \cos \phi =3 cos(0.841)\approx 2$
Again, we retrieve our original ordered triple (2,1,2) by rounding.

We can also write the transformation from cylindrical coordinates to spherical coordinates:

$\rho=\sqrt{r^2+z^2}$
$\tan \phi = z/r$
$\theta = \theta$.
[Click to show example]
[Click to hide example]
Beginning with the point (3,\pi/3,4) in cylindrical coordinates, we see that
$\rho=\sqrt{r^2+z^2}=\sqrt{3^2+4^2}=\sqrt{25}=5$
$\tan \phi = z/r=4/3$
$\rightarrow \phi=\tan^{-1}(.75)=0.644$
$\theta = \pi/3$.
So the point in spherical coordinates is (5,\pi/3,0.644).

Finally, the transformation from spherical to cylindrical coordinates is given by

$r=\rho \sin \phi$
$\theta =\theta$
$z=\rho \cos \phi$.
[Click to show example]
[Click to hide example]
For one last example, we will take our previous result and transform it back to cylindrical coordinates
$r=\rho \sin \theta=5 \sin (0.644)\approx 3$
$\theta =\pi/3$
$z=\rho \cos \theta5 \cos (0.644)\approx 4$
Therefore the point in cylindrical coordinates is (3, \pi/3, 4) as expected.

Interactive Demonstration

Change of Coordinate Systems Applet

Future Ideas for this Page

• add examples of transformations between three dimensional coordinate systems.

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Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.

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