A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Jaap vd Heide
Date: 2015 Nov 12, 00:11 -0800
I will get back on it in more detail tonight (CET), but what I did is basically constructing the haversine using a unit circle like you can do for the sine and cosine (see https://en.wikipedia.org/wiki/Trigonometry under "extending the definitions"). You might recognise that when cos(a) ranges from 1 (at a=0) to -1 (at a=180), 1-cos(a) will range from 0 (at a=0) to 2 (at a=180).
Dividing it all by 2, so (1-cos(a))/2 then ranges from 0 (at a=0) to 1 (at a=180). (1-cos(a))/2 = haversine(a) by definition.
The idea of solving the haversine form of the spherical law of cosines I got from Erik de Man's nautical pages. He has a worksheet for graphically reducing the sight with the sine/cosine form of the equation.
My first mentioning of graphically reducing the sight as I propose was in a post on condensed tables. Not a whole lot more than referring to the model I had put on Geogebra.
I don't think I can do a better job than John Karl did in "Celestial navigation in the GPS age" in showing it all comes down to applying the spherical law of cosines to the navigational triangle.