Many scientific and technical applications use polar coordinates in the Euclidean space to solve some of the complex geometric problems.
Let P be the chosen fixed point in the plane. The half-line $\overrightarrow{o}$ with the start point P and a determined anti-clockwise revolution about P in the plane form the polar coordinate system $\left(P\mathrm{,}\phantom{\rule{2mm}{2mm}}\overrightarrow{o}\mathrm{,}\phantom{\rule{2mm}{2mm}}\varphi \right)$. Point P is called the pole - origin of the coordinate system, half-line $\overrightarrow{o}$ is the polar axis of the polar coordinate system.
Figure 1: Polar coordinates |
Choosing the measurement unit, any point M in the plane can be attached an ordered pair of real numbers $M=\left(\rho \mathrm{,}\phantom{\rule{mediummathspace}{0.2em}}\varphi \right)$ with the clear geometric interpretation illustrated in the Figure 1.
The ordered pair of real numbers $\left(\rho \mathrm{,}\varphi \right)$ form polar coordinates of the point, number $\rho $ is called modul, number $\varphi =[0,2\pi )$ or $\varphi =(-\pi ,\pi ]$ is called the polar angle.
Let the Cartesian coordinate system (O, x, y) and the polar coordinate system $\left(P\mathrm{,}\phantom{\rule{2mm}{2mm}}\overrightarrow{o}\mathrm{,}\phantom{\rule{2mm}{2mm}}\varphi \right)$ be given in the Euclidean plane ${\mathbf{E}}^{2}\left(R\right)$.
These two coordinate systems are called related, if
and choosing one the other coordinate system is determined uniqely.
Figure 2: Related coordinate systems |
If $({x}_{M}\mathrm{,}{y}_{M})$ are Cartesian coordinates and $\left(\rho \mathrm{,}\phantom{\rule{mediummathspace}{0.2em}}\varphi \right)$ are polar coordinates of the point M, their relation can be expressed by equations
$${x}_{M}=\rho \mathrm{cos}\phantom{\rule{mediummathspace}{0.2em}}\varphi $$ |
$${y}_{M}=\rho \mathrm{sin}\phantom{\rule{mediummathspace}{0.2em}}\varphi $$ |
and because
$${x}_{M}^{2}+{y}_{M}^{2}\ne 0$$ |
$$\rho =\sqrt{{x}_{M}^{2}+{y}_{M}^{2}}$$ |
also by equations
$$\mathrm{cos}\phantom{\rule{mediummathspace}{0.2em}}\varphi =\frac{{x}_{M}}{\sqrt{{x}_{M}^{2}+{y}_{M}^{2}}}$$ |
$$\mathrm{sin}\phantom{\rule{mediummathspace}{0.2em}}\varphi =\frac{{y}_{M}}{\sqrt{{x}_{M}^{2}+{y}_{M}^{2}}}$$ |
If $\rho =0$ and thus ${x}_{M}={y}_{M}=0$, the polar angle is not defined by the above equations and the polar coordinates of the point are $P=\left(\mathrm{0,}\phantom{\rule{mediummathspace}{0.2em}}\varphi \right)$, where $\varphi \phantom{\rule{2mm}{2mm}}$is an arbitrary number.