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April 1

Great circle distance between three points

If I know the great circle distance between Point A and Points B and C, is it possible to calculate the distance between Point B and Point C? For example, the distance between Albany, NY and Annapolis, MD is 291 miles, and the distance between Albany and Atlanta, GA is 842 miles. Can I use these numbers to calculate the distance between Annapolis and Atlanta? - Presidentman talk · contribs (Talkback) 21:45, 1 April 2020 (UTC)[reply]

@Presidentman: No. The triangle inequality is the most you can say. When the Earth's curvature is negligible (which is not the case for your example), the law of cosines describes the relationship.--Jasper Deng (talk) 21:50, 1 April 2020 (UTC)[reply]

We have Spherical law of cosines if you know the angle between the two known arcs. -- ToE 05:42, 2 April 2020 (UTC)[reply]

Also relevant is Solution of triangles. -- ToE 08:42, 2 April 2020 (UTC)[reply]

A simple example can illustrate that this is not possible in general (assuming no points coincide). Consider the following four points A, B, C1 and C2. Point A is the North Pole, The other three are all on the equator. Point B is Makoua, in Africa. Point C1 is Butembo, also in Africa. Point C2 is Ciudad Mitad del Mundo, in South America. The distances from A to B and C1 are the same as the distances from A to B and C2. If it was possible to find the third distances from these distances, then the distance between B and C1 would be the same as the distance between B and C2. But the distance between Makoua and Ciudad Mitad del Mundo is more than six times that between Makoua and Butembo.
Another way of thinking about it: on a small scale, a sphere seems flat. If this was possible on a sphere, it should also be possible on a plane. There is one class of special cases on the sphere where the solution is possible, which is when A is antipodal to one of B and C. Let d(A,B) stand for the distance between A and B, and similarly for the other combinations of two points. If the radius of the sphere is R, and d(A,B) = πR (so A and B form an antipodal pair), then d(B,C) = πR − d(A,C). --Lambiam 06:47, 2 April 2020 (UTC)[reply]

To state this in other words, if it's not a trivial solution, you need three pieces of information (regardless if you're on a plane or a sphere) and your two distances are only two. Any of the angles between the arcs will do for the third piece, but you must have a third piece. 93.136.17.76 (talk) 10:40, 5 April 2020 (UTC)[reply]