Polar coordinates

For an extension to three dimensions, see spherical polar coordinates.

CC Image
Two dimensional polar coordinates r and θ of vector $\scriptstyle {\vec {\mathbf {r} }}$

In mathematics and physics, polar coordinates are two numbers—a distance and an angle—that specify the position of a point on a plane.

In their classical ("pre-vector") definition, polar coordinates give the position of a point P with respect to a given point O (the pole) and a given line (the polar axis) through O. One real number (r ) gives the distance of P to O and another number (θ) gives the angle of the line O—P with the polar axis. Given r and θ, one determines P by constructing a circle of radius r with O as origin, and a line with angle θ measured counterclockwise from the polar axis. The point P is on the intersection of the circle and the line.

In modern vector language one identifies the plane with a real Euclidean space $\scriptstyle \mathbb {R} ^{2}$ that has a Cartesian coordinate system. The crossing of the Cartesian axes is on the pole, that is, O is the origin of the Cartesian system and the polar axis is identified with the x-axis of the Cartesian system. The line O—P is generated by the vector

${\overrightarrow {OP}}\equiv {\vec {\mathbf {r} }}.$

Hence we obtain the figure on the right where $\scriptstyle {\vec {\mathbf {r} }}$ is the position vector of the point P.

Algebraic definition

The polar coordinates r and θ are related to the Cartesian coordinates x and y through

${\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}\\x&=r\cos \theta \\y&=r\sin \theta ,\\\end{aligned}}$

so that for r ≠ 0,

$\theta ={\begin{cases}\arccos(x/r)&{\hbox{ if }}y\geq 0\\360^{0}-\arccos(x/r)&{\hbox{ if }}y<0.\\\end{cases}}$

Bounds on the coordinates are: r ≥ 0 and 0 ≤ θ < 360⁰. Coordinate lines are: the circle (fixed r, all θ) and a half-line from the origin (fixed direction θ all r). The slope of the half-line is tanθ = y/x.

Surface element

The infinitesimal surface element in polar coordinates is

$dA=J\,dr\,d\theta .$

The Jacobian J is the determinant

$J={\frac {\partial (x,y)}{\partial (r,\theta )}}={\begin{vmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \\\end{vmatrix}}=r\cos ^{2}\theta +r\sin ^{2}\theta =r.$

Example: the area A of a circle of radius R is given by

$A=\int _{0}^{2\pi }\int _{0}^{R}r\,dr\,d\theta =\pi R^{2}.$