Change of Coordinate Systems
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:*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. | :*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. | ||
- | + | ====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> | ||
+ | |||
+ | Now 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>. | ||
+ | |||
+ | 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. | ||
===Interactive Demonstration=== | ===Interactive Demonstration=== |
Revision as of 11:03, 12 October 2009
Change of Coordinates |
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Change of Coordinates
- The same object, here a disk, can look completely different depending on which coordinate system is used.
Contents |
Basic Description
It is a common practice in mathematics to use different coordinate systems to solve different problems. An example of a switch between coordinate systems follows: suppose we take a set of points in regular x-y Cartesian Coordinates, represented by ordered pairs such as (1,2), then multiply their x-components by two, meaning (1,2) in the old coordinates is matched with (2,2) in the new coordinates.Under this transformation, a set of points would 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). 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
Some of these mappings can be neatly represented by vectors and matrices, in the form
Where is the coordinate vector of our point in the original coordinate system and 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
As can be easily verified.
The ellipse that is tilted relative to the coordinate axes is created by a combination of rotation and stretching, represented by the matrix
Some very useful mappings cannot be represented in matrix form, such as mapping points from Cartesian Coordinates to Polar Coordinates. Such a mapping, as shown in this page's main image, can map a disk to a rectangle. Each origin-centered ring in the disk consists of points at constant distance from the origin and angles ranging from 0 to . 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
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 coordinates, where each coordinate is a distance on a coordinate axis.
- Cylindrical Coordinates use , where are the same as two-dimensional polar coordinates and z is distance from the x-y plane.
- Spherical Coordinates use , where is the distance from the origin, is rotation from the positive x-axis as in polar coordinates, and is rotation from the positive z-axis.
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.
Now we can calculate the cylindrical coordinates for the point given by (1,2,3) in Cartesian coordinates.
- radians
So we have the point in cylindrical coordinates.
The conversion from cylindrical coordinates to Cartesian coordinates is given by
- .
Now, we can convert the point in cylindrical coordinates back to Cartesian coordinates.
- .
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.
Interactive Demonstration
Future Ideas for this Page
- add examples of transformations between three dimensional coordinate systems.
Teaching Materials
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