Polar coordinates

File:2D polar coordinates .png

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

In mathematics and physics, polar coordinates give the position of a vector $\scriptstyle {\vec {\mathbf {r} }}$ in two-dimensional real space $\scriptstyle \mathbb {R} ^{2}$. A Cartesian system of two orthogonal axes is presupposed. One number (r) gives the length of the vector and the other number (θ) gives the angle of the vector with the x-axis of the Cartesian system (measured in the direction of the positive y-axis).

Definition

The polar coordinates 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,

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