Spherical Coordinates — Question 8

Question 8

Two points have spherical coordinates $(\rho_1,\theta_1,\phi_1)$ and $(\rho_2,\theta_2,\phi_2)$. Derive a formula for the square of their distance using their central angle.

Question 8 – Solution

Strategy Write the spherical unit direction vectors and use the dot-product form of the law of cosines.

Direction dot product If $\gamma$ is the angle between the radial directions, then $$\cos\gamma=\cos\phi_1\cos\phi_2+ \sin\phi_1\sin\phi_2\cos(\theta_1-\theta_2).$$

Distance With position vectors $\vec p_1,\vec p_2$, $$d^2=\|\vec p_1-\vec p_2\|^2 =\rho_1^2+\rho_2^2-2\rho_1\rho_2\cos\gamma.$$ Therefore $$\boxed{d^2=\rho_1^2+\rho_2^2-2\rho_1\rho_2 \bigl[\cos\phi_1\cos\phi_2+\sin\phi_1\sin\phi_2\cos(\theta_1-\theta_2)\bigr]}.$$