Change of Coordinate Systems
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|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 circle, 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. | + | |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. |
| - | 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 | + | Under this transformation, a set of points would become stretched in the horizontal x-direction since each point becomes further from the vertical y-axis (except for points originally on the y-axis, which remain on the axis). A set of points that was originally contained in a circle in the old coordinates would be contained by a stretched-out ellipse in the new coordinate system, as shown in this page's main image. |
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| + | Points can even be transferred to a different kind of coordinate system. A common example is mapping rectangular Cartesian Coordinates to '''Polar Coordinates'''. Each point's distance from the origin, R, and angle from the x-axis, <math> \theta </math>, are used as coordinates in the Polar Coordinate system. For example, the point (0,1) in the Cartesian system is a distance 1 from the origin and an angle <math> \pi/2 </math> from the x-axis, giving it polar coordinates <math> (r,\theta) = (1,\pi/2) </math>. Thus a disk in Cartesian Coordinates is mapped to a rectangle in Polar Coordinates: each origin-centered circle consists of points equidistant from the origin with angles from the x-axis ranging from zero to <math> 2\pi </math> radians. Each of these circles is thus mapped to a straight line of length <math> 2\pi </math> in Polar Coordinates. Since the distances from the origin of these circles range from zero to the radius of the disk, a set of lines is created in Polar Coordinates which together form a rectangle. | ||
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|ImageDesc=Points in one space are undergo a transformation of some kind to be mapped to a points in another space. | |ImageDesc=Points in one space are undergo a transformation of some kind to be mapped to a points in another space. | ||
|AuthorName=Brendan John | |AuthorName=Brendan John | ||
Revision as of 13:30, 11 June 2009
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Change of Coordinates
- The same object, here a circle, can look completely different depending on which coordinate system is used.
Basic Description
It is a common practice in mathematics to use different coordinate systems to solve different problems. An example of a switch between coordinate systems follows: suppose we take a set of points in regular x-y Cartesian Coordinates, represented by ordered pairs such as (1,2), then multiply their x-components by two, meaning (1,2) in the old coordinates is matched with (2,2) in the new coordinates.Under this transformation, a set of points would become stretched in the horizontal x-direction since each point becomes further from the vertical y-axis (except for points originally on the y-axis, which remain on the axis). A set of points that was originally contained in a circle in the old coordinates would be contained by a stretched-out ellipse in the new coordinate system, as shown in this page's main image.
Points can even be transferred to a different kind of coordinate system. A common example is mapping rectangular Cartesian Coordinates to Polar Coordinates. Each point's distance from the origin, R, and angle from the x-axis, 
, are used as coordinates in the Polar Coordinate system. For example, the point (0,1) in the Cartesian system is a distance 1 from the origin and an angle 
from the x-axis, giving it polar coordinates 
. Thus a disk in Cartesian Coordinates is mapped to a rectangle in Polar Coordinates: each origin-centered circle consists of points equidistant from the origin with angles from the x-axis ranging from zero to 
radians. Each of these circles is thus mapped to a straight line of length in Polar Coordinates. Since the distances from the origin of these circles range from zero to the radius of the disk, a set of lines is created in Polar Coordinates which together form a rectangle.
A More Mathematical Explanation
Points in one space are undergo a transformation of some kind to be mapped to a points in another spa [...]
Points in one space are undergo a transformation of some kind to be mapped to a points in another space.
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