# Distance and Midpoint in Spherical Coordinates - Calculator

## Formulas Used in Calculations

Given two points by their spherical coordinates, this calculator calculates the distance between the two points and their midpoint.
Given the spherical coordintes of point $P_1(\rho_1,\theta_1,\phi_1)$ and point $P_2(\rho_2,\theta_2,\phi_2)$, we first convert the coordinates of each point to rectangular coordinates written as $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$
where
$x_1 = \rho_1 \sin \phi_1 \cos \theta_1$ , $y_1 = \rho_1 \sin \phi_1 \sin \theta_1$ , $z_1= \rho_1 \cos \phi_1$
$x_2 = \rho_2 \sin \phi_2 \cos \theta_2$ , $y_2 = \rho_2 \sin \phi_2 \sin \theta_2$ , $z_2= \rho_2 \cos \phi_2$

The distance $d(P_1 P_2)$ between points $P_1$ and $P_2$ is given by
$d(P_1 P_2) = \sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

The rectangular coordinates of the midpoint $M(x,y,z)$ of the segment $P_1 P_2$ are given by
$x = \dfrac{x_1+x_2}{2}$ , $y = \dfrac{y_1+y_2}{2}$ , $z = \dfrac{z_1+z_2}{2}$

Then the rectangular coordinates of the midpoint are converted back to spherical coordinates as follows
$\rho = \sqrt {x^2 + y^2 + z^2}$ , $\tan \theta = \dfrac{y}{x}$ , $\cos \phi = \dfrac{z}{\sqrt {x^2 + y^2 + z^2}}$
with $0 \le \theta \lt 2\pi$ and $0 \le \phi \le \pi$

## Use Calculator to Calculate Distance and Midpoint Between Points Given by Spherical Coordinates

1 - Enter the spherical coordinates $\rho_1$ , $\theta_1$, $\phi_1$ of point $P_1$, and the spherical coordinates $\rho_2$ , $\theta_2$, $\phi_2$ of point $P_2$, selecting the desired units for the angles, and press the button "Calculate". You may also change the number of decimal places as needed; it has to be a positive integer.

 $\rho_1 =$ 1 $\theta1 =$ 45 degrees radians $\phi_1 =$ 45 degrees radians $\rho_2 =$ 2 $\theta2 =$ 90 degrees radians $\phi_2 =$ 30 degrees radians
Number of Decimal Places =

Distance =
$\rho =$
$\theta =$   Radians
$\theta =$   Degrees
$\phi =$   Radians
$\phi =$   Degrees