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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.

\( \rho_1 = \)
\( \theta1 = \)
\( \phi_1 = \)
\( \rho_2 = \)
\( \theta2 = \)
\( \phi_2 = \)
Number of Decimal Places =



Distance =
\( \rho = \)
\( \theta = \)   Radians
\( \theta = \)   Degrees
\( \phi = \)   Radians
\( \phi = \)   Degrees


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