student:mathematics:coordinate-systems-2d [JMU CS Wiki]

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===== Coordinate Systems for the Plane =====



==== Introduction ====

//Coordinates// are quantities that designate the position of a point  in relation to a given reference frame. The quantities can be linear and/or  angular, and we will consider  both here. We will limit our discussion to points on the plane.


==== Rectangular Coordinates ====


In a //rectangular coordinate// system the reference frame consists  of an //origin//, a //basis// (more commonly called the  coordinate axes), and a directional convention. 



=== Traditional Cartesian Coordinates ===


In traditional Cartesian coordinates the origin is at located at  the "center" of the plane, the basis consists of the horizontal  and vertical axis, and positive values are in the "north east"  quadrant (i.e., horizontal coordinates increase from left to right  and vertical coordinates increas from down to up). This is illustrated  in the following figure.

{{cartesian-coordinates.gif}}

The horizontal coordinate is traditionally denoted by \(x\)  and the vertical coordinate is traditionally denoted by \(y\).


=== Screen Coordinates ===


Because of the way old hardware worked, most computer  screens/displays do not use Cartesian coordinates. Instead, they  use a rectangular system in which the origin is at the upper-left  corner, horizontal coordinates increase from left to right, and  vertical coordinates increase from up to down (and, as a result,  there are no negative coordinates). This is illustrated in the  following figure.

{{coordinates.gif}}

==== Polar Coordinates ====


In a //polar coordinate// system the reference frame consists  of and origin located at the "center" of the plane, an angular  rotation around origin (in which 0 is to the right the angle  increases counter-clockwise), and a linear distance from the  origin. Especially in the sciences and engineering, the angular  quantity is commonly called the //direction// and the linear  quantity is commonly called the  //magnitude//. This is illustrated in the following figure.


{{polar-coordinates.gif}}

While the angular quantity can be measured in radians or degrees, it is most commonly measured in radians.


==== Converting between Polar and Cartesian Coordinates ====


One often needs to convert polar coordinates to Cartesian coordinates  and //vice versa//. Fortunately, the process is straightforward  and only involves some simple trigonometry. 


In the following figure, the point \(p\) can be thought of  as either having the Cartesian coordinates \((p_1, p_2)\) or  the polar coordinates \((\theta, d)\). When converting from  polar coordinates to Cartesian coordinates we have \((\theta, d)\)  and must calculate \((p_1, p_2)\). On the other hand, when converting  from Cartesian coordinates to polar coordinates we have \((p_1, p_2)\)  and must find \((\theta, d)\).

{{cartesian-to-polar.gif}}

Given \((\theta, d)\) it is clear from the figure that  \(\cos(\theta) = p_{1} / d\) and, hence, that  \(p_{1} = d \cos(\theta)\). Similarly, it is clear from the  figure that \(\sin(\theta) = p_{2} / d\) and, hence, that  \(p_{2} = d \sin(\theta)\).

Given \((p_1, p_2)\), on the other hand, it is clear that  \(\tan(\theta) = p_2 / p_1\) for \(p_1 \gt 0\) and,  hence, that \(\theta = \tan^{-1}(p_{2} / p_{1})\). In  addition, observe that \(d\) is the length of the hypotenuse  of the right triangle formed by \((p_1, p_2)\). So, it  follows that \(d = \sqrt(p_1^2 + p_2^2)\).
student/mathematics/coordinate-systems-2d.txt · Last modified: 2018/08/10 15:21 by bernstdh
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