Re: spherical geometry

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Carl's latest attempt:

> Okay, one last attempt. If I interchange the order of integration I
> now find:
>
> area = s - 2*integral from 0 to s/2 of {dY/sqrt(1+tan^2(A)*sin^2(Y)}
>
> That integral almost looks like an elliptic integral, except for the
> sign in front of the trig fn.  It has the correct (trivial) limiting
> values for A = 0 and 90 degrees. Carl

Sorry, but your formula *doesn't* have the correct limiting value for
small s.  In the small s limit the expression above approaches

area = (s^3)/8

(which is dimensionally impossible).  But the correct limiting
formula for the small s area is

area = (sqrt(3)/4)*s^2 .

Also, when the side length s is small A *does* not go to zero; it
goes to 60 deg (whose squared tangent is 3).

David Bowman