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*From*: "Carl E. Mungan" <mungan@USNA.EDU>*Date*: Thu, 19 Aug 2004 14:44:00 -0400

I *agree* with Carl's answer for the Guantanamo Bay location.

Great. Here then is my method. Let the two points on the earth's

surface have (latitude,longitude) of (T1,P1) and (T2,P2) with

north/east positive (ie. T=theta, P=phi). They lie on a great circle

like in my PDF document *except* that this circle no longer crosses

the equator at P=0. Instead it is shifted over to say P=P0. But if we

replace P in my document by deltaP=P-P0, then all the equations will

work as before. In particular, Eq. (3) divided by (2) gives the

constant tilt angle A in terms of T and deltaP. Write this ratio down

twice, once for each set of coordinates, and equate them. That gives

an equation for P0, which can easily be solved to find for the Gitmo

problem:

P0 = 73.625 or -106.375 degrees.

Now that we've found the great circle on which the two points lie, we

simply need to find the angle (call it B say) between any point

(T,deltaP) on this great circle and east. An easy way to do this is

to note that cos(B) = dot product between a unit vector tangent to

the great circle and a unit vector pointing east. Both of these unit

vectors lie in the plane perpendicular to (x,y,z) given by Eqs.

(1)-(3) in my PDF. The east unit vector is therefore

(-y,x,0)/sqrt(x^2+y^2). The tangential unit vector is seen from the

diagram in my PDF to be (-sin G, cos G cos A, cos G sin A) where

G=gamma and A=alpha are the "great circle" coordinates of the point.

After some minor cleanup, this becomes:

sec(B)=sqrt[1+sin^2(T)*cot^2(P-P0)]

Plug in the given values of T1 and P1 with the above value of P0 to

get the desired B. Carl

ps: Thanks to John Denker for going to the trouble to very nicely

draw and re-explain the concept of transporting a vector around an

area on a sphere.

--

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/

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