# NavList:

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

**Re: Applications of Complex Analysis to Celestial Navigation**

**From:**Robin Stuart

**Date:**2009 Oct 26, 16:53 -0700

John, You may have seen the paper by now but if not let me explain that the method does not make use trigonometric functions with complex arguments but just simple arithmetic operations on complex numbers. In a certain sense it is analogous to the use of complex numbers in AC circuit analysis. A point on the surface of a sphere is identified with a complex number by stereographic projection. A single complex number encapsulates both of the angular coordinates (e.g. latitude/longitude, declination/GHA or altitude/azimuth). If z is a complex number representing the declination & GHA of a celestial body and w is a complex number representing its altitude & azimuth then the two are related by w = ( a * z + b ) / ( -Conjugate[b] * z + Conjugate[a] ) where a and b are complex numbers that depend only on the latitude and longitude of the observer. Altitude and azimuth are computed simultaneously by a single equation. As far as Andres' referring to quaternions as a special case of tensor calculus, I took that to mean that the algebra of quaternions is isomorphic to that of the Pauli matrices. I would hesitate to place quaternions on the same level as complex numbers however. The latter, of course, give rise to the whole field of complex function analysis which is not really replicated for quaternions, Robin Stuart --~--~---------~--~----~------------~-------~--~----~ NavList message boards: www.fer3.com/arc Or post by email to: NavList@fer3.com To unsubscribe, email NavList+unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---