Change of Coordinate Systems

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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 th [...]

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.

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. 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 each coordinate is a distance on a coordinate axis.
  • 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.
  • 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.

Transformations between these three coordinate systems exist just as a transformation exists between polar and Cartesian Coordinates.




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