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# Polar coordinates

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

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 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

Hence we obtain the figure on the right where 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

so that for *r* ≠ 0,

Bounds on the coordinates are: *r* ≥ 0 and 0 ≤ θ < 360^{0}.
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

The Jacobian *J* is the determinant

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