Polar coordinates


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



In mathematics and physics, polar coordinates are two numbers&mdash;a distance and an angle&mdash;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 (&theta;)  gives the angle of the line  O&mdash;P with the polar axis. Given r and &theta;, one determines P by constructing  a circle of radius r with O as origin, and  a line with  angle &theta; 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&mdash;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 &theta; are related to the Cartesian coordinates x and y through

\begin{align} r &=  \sqrt{x^2+y^2} \\ x &= r\cos\theta \\ y &= r \sin\theta, \\ \end{align} $$ so that for r &ne; 0,

\theta = \begin{cases} \arccos(x/r)        & \hbox{ if } y \ge 0 \\ 360^0 - \arccos(x/r) & \hbox{ if } y < 0 .\\ \end{cases} $$ Bounds on the coordinates are: r &ge; 0 and 0 &le; &theta; < 3600. Coordinate lines are: the circle (fixed r, all &theta;) and a half-line from the origin (fixed direction &theta; all r). The slope of the half-line is tan&theta; = 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. $$