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Flight Path

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

earth 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 $2d$ (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 $\theta$ radians where $$\sin \theta = {d\over R}.$$ Calculate the arc length $l$.