A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2020 May 21, 08:47 -0700
It may be a bit late now but in order to determine whether a isoazimuth is a small circle I decided to construct a counter example. The plan was to obtain 4 points lying on an isoazimuth. Choosing 3 of those points calculate the centre of the small circle on which they fall. Then choose a different combination of 3 out of the 4 points and calculate the centre on the circle that those fall on. If the centres are different then an isoazimuth is not a small circle.
Circles great and small are conveniently handled analytically by stereographic projection onto the complex plane as described here. Circles on the sphere map to circles on the plane. Moreover since the Littrow projection is a simple analytic function of the stereographic projection this offers a consistent approach for the treatment of equal altitude circles and isoazimuths. In an earlier post by Bill Lionheart provided a link to a useful article which stated "If the complex z-plane is the Littrow map and the w-plane is the polar stereographic map, then z = w +1/w". After some time wrestling with this and trying to make it behave I realized there is a typo. It should read "z = w -1/w" and that makes all the difference.
In any case here's the counter example generated by appealing to the properties of the Littrow projection
Assume the GP of the body being observed is at 16°46.8'N 45°51.1'E. An observer at any of the points
P1 = 10.306896°N 55.375875°E
P2 = 0.248451°S 70.71412°E
P3 = 25.433371°S 95.262638°E
P4 = 48.38593°S 102.746085°E
The points P1, P2 and P4 all lie on a small circle with centre or pole at
c124 = 37.001705°S 28.211937°E
this can be checked calculating their distances from the centre and seeing that they are all the same.
The points P1, P2 and P3 all lie on a small circle with centre or pole at
c123 = 43.700701°S 18.268654°E
These centres are not the same hence isoazimuths are not in general small circles.
Under stereographic projection onto the complex plane isoazimuths appear to satisfy 3rd order equations. It may still be possible to calculate the intersection of 2 isoazimuths or an isoazimuth with a small circle. This would be analogous to the algebraic solution of the double altitude problem discussed here.