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*From*: David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>*Date*: Thu, 19 Aug 2004 11:56:23 -0400

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

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