You need to show that if a place has (latitude, longitude) =

(p, q)

then its coordinates are

(R \cos p \cos q, R \cos p \sin q, R \sin p) .

Steps in the calculation

We need the distance $l$ on the surface between A and B.

First calculate the three-dimensional coordinates of A and B from the latitude and longitude of the two points.

Then calculate the distance $2 d$ (imagine a tunnel straight through the earth from A to B) using Pythagoras Theorem.

Use $R$ , the radius of the Earth, and $d$ to find the angle $θ$ radians where

$\sin θ = \frac{d}{R} .$

Calculate the arc length $l$ .