Polar Coordinate System - Converting Between Polar and Cartesian Coordinates

Converting Between Polar and Cartesian Coordinates

The two polar coordinates r and θ can be converted to the two Cartesian coordinates x and y by using the trigonometric functions sine and cosine:

The Cartesian coordinates x and y can be converted to polar coordinates r and θ with r ≥ 0 and θ in the interval (−π, π] by:

(as in the Pythagorean theorem), and
(where atan2 is a common variation on the arctangent function that takes into account the quadrant)

or

\theta = \begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0\\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\ \frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\ -\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\ 0 & \mbox{if } x = 0 \mbox{ and } y = 0 \end{cases}

In degrees this would be from −180° to 180°. If desired an angle in the range [0, 2π) may be obtained by adding 2π to the value if it is negative. The zero angle at the origin where x and y are both zero is just a convenient value that is often chosen.

The value of θ above is the principal value of the complex number function arg applied to x+iy, except that arg does not define a value at the origin.

Read more about this topic:  Polar Coordinate System

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