We had a modest amount of time to discuss what Van Brummelen calls “The Locality Principle.” His presentation is on pp. 79-80. The aim is to go a bit further here.
We have been discussing triangles on the sphere, and we have made the radius of our sphere 1. Let’s think about what this means for an enormous sphere, like the Earth, with radius R.
See worksheet on Triangles in Deep Springs Valley
(1) Expressed in the “natural” units of Earth-radii, the distances in Deep Springs valley are tiny. We are essentially expressing length in radians when we do this. (2) Even when converted to degrees which involves multiplying by 180/π, the angles are very small. (3) Of course, although the sides are tiny compared to the Earth’s radius, the angles of the triangle on our worksheet are not in any sense small. (4) Our example using the Pythagorean theorem got c = 7500ft. Using cos c = cos a * cos b, and asking Mathematica to work to 20 decimal places, we got c = 7499.9999629129365648ft. (5) As you can see, it makes extraordinarily little difference to treat a triangle involving the main campus and Henderson as a spherical triangle.
A systematic expansion in powers of a/R, b/R, and c/R is possible for every one of our 10 equations. The first non-trivial terms in the expansions yields familiar equations for triangles in the plane. For anyone who has had derivative calculus (integral calculus is not necessary), we can show you Taylor series (aka Maclaurin series) as a special, more advanced, topic. Taylor series are the foundation of many systematic expansions, which we truncate — often at the first non-trivial terms — in order to both (a) discover meaningful approximations and limiting cases that build intuition and provide sanity checks on complex derivations, and (b) make intractable problems tractable.
As an example, on Monday, April 11, we recovered the planar Law of Cosines from the spherical Law of Cosines by expanding to the spherical Law of Cosines in powers of a/R, b/R, and c/R.