Change of Coordinate Systems

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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,  \theta , 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  \pi/2 from the x-axis, giving it polar coordinates  (r,\theta) = (1,\pi/2) . 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  2\pi radians. Each of these circles is thus mapped to a straight line of length  2\pi 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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