Re: spherical geometry

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Ludwik's request:

> I am trying to address Carl's problem in my own way but I need help.
>
> Consider a point A located on a sphere of unit radius. The polar
> axis of the sphere, z on my picture, is vertical. The two spherical
> coordinates of A are TET (polar angle) and PHI (azimuthal angle).
>
> Another polar axis, z', is chosen. Its orientation, in the old
> frame, is specified by ALPHA (polar) and BETA (azimuthal). How are
> new polar coordinates, TET' and PHI',  expressed in terms of old
> polar coordinates? I realize that the transformation is not as
> simple as for xy and x'y'.
>
> Ludwik Kowalski

Ludwik, did you look at the .pdf document that Carl posted at
http://usna.edu/Users/physics/mungan/Scholarship/Triangle.pdf ?
It clearly explains exactly this situation and even has a diagram
that nicely illustrates things.  (You don't even have to learn
Geometric/Clifford Algebra to follow it.)

However, as Leigh originally intimated, and as Carl subsequently
discovered, the simple proof of Girard's theorem is even simpler
than Carl's exposition and requires no calculus or trigonometry to
relate the area to the interior angle.  Also, John's formulation
exploiting the proportionality of the excess turning angle to the
enclosed area is also much simpler (and makes simple direct use of
the fact that a sphere is a space of *constant* curvature).

David Bowman