# Parametric Equations and Polar Coordinates

### Contents

#### Polar Coordinates

The polar coordinate system consists of a pole and a polar axis. The pole is a fixed point, and the polar axis is a directed ray whose endpoint is the pole. Every point in the plane of the polar axis can be specified according to two coordinates: r, the distance between the point and the pole, and θ, the angle between the polar axis and the ray containing the point whose endpoint is also the pole.

Figure %: The polar coordinate system
The distance r and the angle θ are both directed--meaning that they represent the distance and angle in a given direction. It is possible, therefore to have negative values for both r and θ. However, we typically avoid points with negative r, since they could just as easily be specified by adding Π (or 180o) to θ. Similarly, we typically ask that θ be in the range 0≤θ < 2Π, since there is always some θ in this range corresponding to our point. This doesn't eliminate all ambiguity, however; the pole can still be specified by (0, θ) for any angle θ. But it is true that any other point can be described uniquely with these conventions.

To convert equations between polar coordinates and rectangular coordinates, consider the following diagram:

Figure %: The x and y coordinates in the polar coordinate system
See that sin(θ) = , and cos(θ) = .

To convert from rectangular to polar coordinates, use the following equations: x = r cos(θ), y = r sin(θ). To convert from polar to rectangular coordinates, use these equations: r = sqrtx2+y2, θ = arctan().