Polar coordinates

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This editable Main Article is under development and subject to a disclaimer.

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

CC Image
Two dimensional polar coordinates r and θ of vector  r -> {\displaystyle \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 R 2 {\displaystyle \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

O P -> ≡ r -> . {\displaystyle {\overrightarrow {OP}}\equiv {\vec {\mathbf {r} }}.}

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

## Algebraic definition[edit]

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

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

so that for r ≠ 0,

θ = { arccos ⁡ ( x / r ) if y ≥ 0 360 0 − arccos ⁡ ( x / r ) if y < 0. {\displaystyle \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[edit]

The infinitesimal surface element in polar coordinates is

d A = J d r d θ . {\displaystyle dA=J\,dr\,d\theta .}

The Jacobian J is the determinant

J = ∂ ( x , y ) ∂ ( r , θ ) = | cos ⁡ θ − r sin ⁡ θ sin ⁡ θ r cos ⁡ θ | = r cos 2 ⁡ θ + r sin 2 ⁡ θ = r . {\displaystyle 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 = ∫ 0 2 π ∫ 0 R r d r d θ = π R 2 . {\displaystyle A=\int _{0}^{2\pi }\int _{0}^{R}r\,dr\,d\theta =\pi R^{2}.}