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I am looking for the the method of constructing the midpoint of two points in spherical geometry. The only tools allowed for the construction are a pair of spherical compasses and a spherical ruler.

In Eclidean geometry constructing the midpoint is relatively easy. We are looking for the midpoint of points A and B. We construct two circles on A and B with the radiuses of AB. Then we construct two straight lines. One is through the two intersections of the two circles and one is through A and B. The intersection of these two lines will give the midpoint of A and B.

It is clear that the Euclidean method of construction does not work in Spherical geometry. The circles do not intersect when the distance of our two points exceeds 120°. There is also no solution when their distance is exactly 90°.

How would you construct the midpoint of two points in spherical geomerty?

Thank you

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Draw a great circle $L$ between $A$ and $B$. Set your compass to a small distance, say $r$. Mark off distance $r, 2r, 3r, \ldots$ along $L$, moving from $A$ to $B$. Now do the same from $B$ to $A$. Count to find the middle interval and subdivide that.

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Thank you for your answer. How would you choose r if you could use only the distances of points? –  ererer Dec 18 '10 at 12:35
    
Suppose that A, B are not antipodal. Let M be the circle through A with center B. Then M and L meet in another point, C. Use the shorter arc of L between A and C to define r. Of course, this won't work when A, B are antipodal. In this case you have to make a choice of great circle L! –  Sam Nead Dec 18 '10 at 17:19
    
One more remark -- making arbitrary choices is "allowed" in Euclidean geometry. See Propositions I.5, I.9, I.11, I.12, I.23, I.31, ... in Euclid's Elements. Not all of these choices can be eliminated. I.31 is a particularly nice case to consider. –  Sam Nead Dec 18 '10 at 17:23
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