# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Using any star for a lunar**

**From:**Bill B

**Date:**2005 Apr 7, 00:18 -0500

> Mark three points around the equator. Point A is at 0deg West, point B is > at 120deg West, point C is at 240deg West. Join A to B, B to C, and C to A > again, with great circles, going Westerly each time. Then you have three > vertexes, each subtending a 180deg angle. The resulting triangle divides > the circle into two equal halves > > You might object that with such a 180deg angle, then each vertex has become > a straight line, not a corner. That's true, because it's a limiting case. > Think about it, if you prefer, when the angle at A, B, and C is not quite > (but as near as dammit) equal to 180, so there's a VERY obtuse angle at > each corner. Then increase these angles, very slightly. George, See your point(s). Had not considered that case, C to A as the smallest GCD; but rather joined my three arbitrary points (C to A) with the greater GCD. > > This business of the "spherical excess" is quite new to me, but it seems to > work. If we add the three "angles", each 180, we get 540 deg. The spherical > excess over a tiny triangle, in which the angles always sum to 180deg, is > therefore 360 deg, or 2pi radians. Multiply this by r squared, and we get 2 > pi r-squared, which is indeed the area of the half-sphere that the > "triangle" embraces. Today, I've learned something new... Ain't life grand? Been doing dumb human tricks with spherical trig today. Once I got got the bigger picture, it occurred to me if swap out AP Lat and GP Lat (declination) and vise versa in the Z formula(s) I could obtain the AP, GP, pole angle. Add those up (-180d, pi radians) and I can calculate area in the triangle. I am dumbfounded as to how to use the area inside the triangle--other than to impress sailing friends whose eyes glaze over when I mention cel nav with the number of square miles of water included in the triangle--but betting one the list mentors has a practical application. Bill