From: m9305474@student.anu.edu.au (John McLaughlin)
Newsgroups: comp.graphics.algorithms,alt.3d,sci.math
Subject: Re: Object rotation and navigation on a sphere
Date: Mon, 01 May 1995 10:53:48 +1000

In article <3nu50r$s48@zeus.ptltd.com>, tom_roden@ptltd.com (Tom Roden) wrote:

> greetings,
> 
> I have been doing some work with different ways to express an 
> object's rotation other than the standard 3x3 or 4x4 matrix.  
> I have found an equivalence between this problem and that of 
> navigation on a sphere.  Thus,  am trying to locate equations 
> pertaining to navigation on a sphere.
> 
> Given current longitude, latitude, heading, and distance, I would 
> like to determine resulting longitude, latitude, and heading.  
> Distance could also be expressed as amount of great circle arc 
> (i.e. distance_in_radians).
> 

I'm not sure how you represent heading here. Is it 2d vector containing
change in longitude and change in latitude? Anyway, I can give you the
following equations describing triangles on a sphere. If the triangle has
internal angles alpha, beta, gamma and the angular length of the sides of
the triangles is a,b,c where a is opposite alpha, b is opposite beta and c
is opposite gamma. Then,

cos(alpha) = ( cos(a) - cos(b)cos(c) ) / (sin(b)sin(c))

cos(a)     = ( cos(alpha) + cos(beta)cos(gamma) ) / (sin(beta)sin(gamma))

Interesting eh?
Permutate (a,b,c) and (alpha,beta,gamma) to get the other 4 equations. You
should be able to derive the navigation problem by using the properties of
the above equations. Form a triangle between the old position, the new
position and the north pole.

-John McLaughlin