Re: spherical geometry

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Carl's rearrangement of his formula:

> First let me simplify my formula to:
>
> tan(B) = sin(T) * cot(P-P0)
>
> Are you happier with this? It allows negative values, although of
> course it still can't distinguish quadrants I/IV from II/III.

At least twice as happy as before.  Actually, the ambiguity your
result still now has is *supposed* to be there.  This is because
there are 2 geodesic paths connecting two arbitrary fixed points
on a sphere going in opposite directions.  Your formula now gives
both the short way *and* the long way around the sphere from the
same formula.

> > If you want another spherical geometry exercise,
> > rederive Clairaut's formula.
> >    http://williams.best.vwh.net/avform.htm#Clairaut
>
> Heck, I'm never going to get anything else done, but it's fun.
> Equation (3) divided by (2) in my PDF gives:
>
> tan(A) = tan(T)/sin(P-P0)
>
> Solve this for (P-P0) and substitute into my bearing formula above
> to get:
>
> cos(T) * cos(B) = cos(A) = constant
>
> which is Clairaut's formula, and certainly looks like a nice way to
> satisfy Hugh's request for a simple method of correcting course
> enroute. Swap cos(B) for sin(H) for Brian's desire to get headings
> relative to north. -Carl

Good work!

David Bowman