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Polar Coordinates

The definition of polar coordinates can be given in many forms, one of the easiest way of defining polar coordinates is given below:

• Polar cordinates are the coordinate system in which each point is represented by a distance from a fixed point called POLE and the angle from a fixed direction. Suppose a point is at a distance of 5 units from pole and at an angle 600 , then it is represented in polar form as (5, 600 )

Here "O" is the POLE and OB is the polar axis, the distance OA is called the radial coordinate (r) and the angle 600 is called the angular coordinate The radial coordinate is often denoted by r, and the angular coordinate by ? or t .Angles in polar notation are generally expressed in either degrees or radians.

• The 2-D polar coordinate system involves the distance from the origin and an azimuth angle. The figure shows the 2-D polar coordinate system, where r is the distance from the origin to point P, and 0 is the azimuth angle measured from the horizontal (X) axis in the counterclockwise direction. Thus, the position of point P is described as (r,0). r & 0 are the 2-D polar coordinates.

What are the uses of polar coordinates?

The polar coordinates are useful in describing the human body motion since the essence of the human body motion is the joint motions. The segments undergo rotations about the joint centers and using the azimuth angle (while r = const) in describing the body motion is more efficient than using the Cartesian coordinates. One thing to note here is that coordinates r & 0 are not the same kind.

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Polar Coordinates to Cartesian Coordinates

Lets take an example of converting polar coordinate into cartesian coordinates. The two polar coordinate r and θ can be converted to the Cartesian coordinates x and y by using the trignometric functions sine and cosine:

$x = r \cos \theta \,$
$y = r \sin \theta, \,$

while the two Cartesian coordinates x and y can be converted to polar coordinate r by

$\theta = \begin{cases} 0 & \mbox{if } x = 0 \mbox{ and } y = 0\\ \arcsin(\frac{y}{r}) & \mbox{if } x \geq 0 \\ -\arcsin(\frac{y}{r}) + \pi & \mbox{if } x < 0\\ \end{cases}$

Cartesian Coordinates to Polar Coordinates

For converting cartesian coordinate to polar coordinate lets take an example. Lets take a point in cartseian coordinates and convert it into polar coordinates. If you have a point in Cartesian Coordinates (x,y) and need it in Polar Coordinates (r,θ), you need to solve a triangle where you know two sides.

What is (12,5) in Polar Coordinates?

We use the pythagorean theorem to find the longest side as

r2= 122 + 52

r = √ (122 + 52)

r = √ (144 + 25) = √ (169) = 13

we use the tangent function to find the angle

tan( θ ) = 5 / 12

θ = tan-1( 5 / 12 ) = 22.6 °

so the cartesian coordinate (12,5) in

polar coordinate is (13,22.60 )

So, to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

r = √ (x2 + y2)

θ = atan( y / x )

 More topics in Polar Coordinates Polar Equations of Lines Graphing Polar Equations Rectangular to Polar Coordinates Circle in Polar Coordinates Spherical Coordinates Parametric Functions Distance Formula for Polar Coordinates
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