Spherical coordinates

Spherical coordinate system in the space is determined by the plane $\pi$, in which the oriented half-line ${\overrightarrow{p}}_1$ with the start point S and the anti-clockwise revolution are chosed, and by the oriented half-line ${\overrightarrow{p}}_2$ with the start point S and perpendicular to $\pi$.

In the spherical coordinate system $\left(S\mathrm{,}\rho \mathrm{,}\varphi \mathrm{,}\zeta \right)$ any point M in the space, which is not on line p coinciding with the half-line ${\overrightarrow{p}}_2$, can be attached a unique triple of real numbers $\rho \mathrm{,}\varphi$ and $\zeta$, with a clear geometric interpretation

Figure 1: Spherical coordinate system

1. $\rho =\left|SM\right|$, so $\rho$ is the distance of points M and S
2. $\varphi =|\sphericalangle \left({\overrightarrow{p}}_1\mathrm{,}\overrightarrow{S{M}_1}\right)|$, thus $\varphi$ is size of the oriented angle with the vertex in point S, the first arm is formed by half-line $\overrightarrow{p_1}$, the second arm by half-line ${\overrightarrow{SM}}_1$, while $M_1$ is the orthographic view of M in the plane $\pi$
3. $\zeta =|\sphericalangle \left({\overrightarrow{p}}_2\mathrm{,}\overrightarrow{SM}\right)|$, thus $\zeta$is the size of angle formed by $\overrightarrow{p_2}$ and $\overrightarrow{SM}$.

An ordered triple of real numbers $\left(\rho \mathrm{,}\varphi \mathrm{,}\zeta \right)$ form spherical coordinates of the point M, for which holds $\rho \in (\mathrm{0,}\infty )$, $\varphi \in [\mathrm{0,}2\pi )$, or $\varphi \in (-\pi \mathrm{,}\pi ]\mathrm{,}\zeta \in [\mathrm{0,}\pi )$.

If point M is located on line p, its spherical coordinates are in the form of a triple

where $\varphi$ is and arbitrary number, and $\zeta =0$ for points M located on half-line $\overrightarrow{p_2}$ , while $\zeta =\pi$ in the opposite case.

Figure 2: Relation between spherical and Cartesian coordinates

Spherical coordinate system $\left(S\mathrm{,}\rho \mathrm{,}\varphi \mathrm{,}\zeta \right)$ is related (conjugate) to the Cartesian orthogonal coordinate system $\left(O\mathrm{,}x\mathrm{,}y\mathrm{,}z\right)$ in the space, iff

1. plane $\pi$ determining the spherical coordinate system $\left(S\mathrm{,}\rho \mathrm{,}\varphi \mathrm{,}\zeta \right)$ coincides with the plane $xy$of the Cartesian coordinate system $\left(O\mathrm{,}x\mathrm{,}y\mathrm{,}z\right)$
2. point S coincides with the origin O and oriented half-line $\overrightarrow{p_1}$ coincides with the positive part of the axis x
3. oriented half-line $\overrightarrow{p_2}$ coincides with the positive part of the axis z.

If $\left(x\mathrm{,}y\mathrm{,}z\right)$ are Cartesian coordinates and $\left(\rho \mathrm{,}\varphi \mathrm{,}\zeta \right)$ are spherical coordinates of point M not located on the coordinate axis z, their relation can be represented by the following equations

and since $\left|S{M}_1\right|=\rho \sin \zeta$ , it is valid

$$x=\rho \cos \varphi \sin \zeta$$

$$y=\rho \sin \varphi \sin \zeta$$

$$z=\rho \cos \zeta$$

from which the spherical coordinates can be expressed

Set of all points in the space with the constant first spherical coordinate $\rho =a>0$ is sphere with centre in the origin of the coordinate system S and radius a.