Re: spherical geometry

["Carl E. Mungan" <mungan@USNA.EDU>]

David Bowman wrote:

> Carl, this integral can be integrated using the substitution to
> the new variable X:
>
> X = arcsin(sin(A)*cos(Y))
>
> the smoke *eventually* clears and the formula for your integral
> boils down to the *correct* expression:
>
> area = 3*A - [pi]

Fantastic! I modified my PDF document to include these suggestions.

Now that I fully understand what the result is, can you (and/or John
Denker) please explain how you knew there would be a simple (linear)
relationship between the area and interior angle of the triangle? I
didn't fully grasp the previous discussion about transporting a
vector around the triangle, even though I see how it works for the
special case of one octant. In other words:

1. How do you know that the rotation of the transported vector is
linearly proportional to the area?

2. How do you know (independently of 1) that the rotation is linearly
proportional to the amount of angle you underwent in jumping from one
great circle to another?
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
      U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu       http://usna.edu/Users/physics/mungan/