Change of Coordinate Systems

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|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.
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|ImageDescElem=It is a common practice in mathematics to use different coordinate systems to solve different problems. In two dimensional space, 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. 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).
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|ImageDescElem=It is a common practice in mathematics to use different coordinate systems to solve different problems. 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.
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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.
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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 by 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>, is used as coordinates in the Polar Coordinate system. 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 angle from the x-axis ranging from zero to <math> 2\pi </math> radians. Each of these circles is thus mapped a straight line of length <math> 2\pi </math> in Polar Coordinates. Since the distance from the origin of these circles ranges from zero to the radius of the disk, a set of lines is created in Polar Coordinates which together form a rectangle.
|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 12:06, 11 June 2009

Image:inprogress.png

Change of Coordinates

The same object, here a circle, can look completely different depending on which coordinate system is used.


Basic Description

It is a common practice in mathematics to use different coordinate systems to solve different problems. 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 by 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,  \theta , is used as coordinates in the Polar Coordinate system. 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 angle from the x-axis ranging from zero to  2\pi radians. Each of these circles is thus mapped a straight line of length  2\pi in Polar Coordinates. Since the distance from the origin of these circles ranges 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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