Re: spherical geometry

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Carl's latest incarnation of his integral form solution
for the area of an equilateral triangle on a unit sphere:

> The corrected integral now becomes:
>
> area = 2*integral from 0 to s/2 of {dY/sqrt(1+cot^2(A)*csc^2(Y)}

Carl, this integral can be integrated using the substitution to
the new variable X:

X = arcsin(sin(A)*cos(Y))

After using some trigonometric identity formulas along with
the relationship that A is determined by s to be:

A = 2*arcsin(sec(s/2)/2) = arccos(tan(s/2)/tan(s)) =
  = arccos(1/(1 + 1/cos(s))) = [other equivalent formulae, too]

the smoke *eventually* clears and the formula for your integral
boils down to the *correct* expression:

area = 6*arcsin(sec(s/2)/2) - [pi] = 3*A - [pi]

The final result is also correct when s is even obtuse although
the integral formulation above manifestly breaks down for obtuse
s values.

I also liked your pdf version.  Nice job!

David Bowman

PS, Your integral version, as messy as it was, was *much* less
messy than my version was to slog through.