What polar coordinates are

Up to now a point of the plane has been located by a pair $(x,y)$ of Cartesian coordinates. There is another natural pair for describing it, particularly handy when the problem has a central symmetry (circles, spirals, trajectories about an origin): the polar coordinates.

Definition — Polar coordinates

Having fixed in the plane a point $O$ (pole) and a ray issuing from $O$ (polar axis, usually coinciding with the positive $x$-semiaxis), the polar coordinates of a point $P\ne O$ are the pair $(\rho,\theta)$ where:

• $\rho = \overline{OP}\ge 0$ is the distance of $P$ from the pole (radius or modulus);
• $\theta\in[0,2\pi)$ (or $(-\pi,\pi]$) is the oriented angle that the ray $OP$ makes with the polar axis (anomaly or argument).

For the pole itself, $\rho = 0$ and $\theta$ is undefined.

The point $P$ is located by the distance $\rho$ from the pole $O$ and by the angle $\theta$ with the polar axis.

Links

Topics: Trigonometry
Concepts: Anomaly · Cartesian coordinates · Polar coordinates · Cartesian plane · Radius