Express spherical coordinates with different centers in terms of each other.

Question

Imagine that you have two spheres with a distance $R$ from one center to the other one. Now, it is well known how one would get the cartesian position vector of each point in sphere 1 by using spherical coordinates with the reference frame being the center of sphere 1. Now, I am looking for an expression that gives me a point written in spherical coordinates with reference point(center 1) in terms of spherical coordinates with reference point(center 2). In other words, I am looking for the relationship between the expression of a point in spherical coordinates with the reference center being the center of sphere number 1 in terms of spherical coordinates with reference to the center of sphere number 2

Answer 1

Let the origin of your first spherical coordinates be the origin of the first Cartesian system being used, and the origin of your second spherical coordinates be at $X_o, Y_o, Z_o$ in the same Cartesian system.

Use spherical polar coordinates, with $\phi$ being the azimuth angle in the $X-Y$ plane, and $\theta$ being the altitude angle measured from the $Z$-axis

Consider a point P, with coordinates $R_1,\phi_1, \theta_1$ in the first spherical coordinate system. Then point P has Cartesian coordinates in the first cartesian system $X_1, Y_1, Z_1$ given by $$X_1=R_1\sin(\theta_1) \cos(\phi_1)$$ $$Y_1=R_1\sin(\theta_1)\sin(\phi_1)$$ $$Z_1=R_1\cos(\theta_1)$$ If we now consider a new Cartesian system parallel to the first and with origin at the origin of the second spherical system, the Cartesian coordinates of P in the second system would be$$X_2=X_1-X_0=R_1\sin(\theta_1) \cos(\phi_1)-X_0$$ $$Y_2=Y_1-Y_0=R_1\sin(\theta_1)\sin(\phi_1)-Y_0$$ $$Z_2=Z_1-Z_0=R_1\cos(\theta_1)-Z_0$$Finally, you can find the polar coordinates of P in the second spherical polar system, $R_2,\phi_2, \theta_2$: $$R_2=\sqrt{X_2^2+Y_2^2+Z_2^2}$$ $$\phi_2=\arctan\frac{Y_2}{X_2}$$ $$\theta_2=\arccos\frac{Z_2}{R_2}$$