For three points on the unit sphere in , suppose that they do not lie on a great circle, then there is a unique great circle on the sphere joining and , and so on. Obviously there are two segments of the great circle joining each pair of points, we can always declare to be the minor segment (i.e. length ) of the great circle joining and . We can also assume to be positively oriented, i.e. . Then we can uniquely define the spherical triangle to be the region on the sphere bounded by the segments and (in this orientation), such that it always lies on the “left hand side” of this oriented closed curve. We are interested to find the area of . After discussing with Ken, Michael, John Ma and Wongting, we have obtained the following results. ☺
With the notations as above, suppose is the interior angle of at the vertex , then
Proof: This can be easily proved by the Gauss-Bonnet theorem, but here I will give a more elementary proof. Suppose is the antipodal point of for any point , it is easy to see that and are congruent (they are mapped to each other by the (orientation reversing) map , which clearly preserves distances), and hence have the same area.
Now consider the region consisting of and , as the arcs and spans an angle of at , we clearly have
So we have the system
But by the previous remark, , so the first equation of the system becomes , and noting that a hemisphere is divided into the four triangles and , so we can add the forth equation into the system:
Summing the first three equations together and subtracting the last one, we can get the result.
The above result is very neat, but given three points , it needs some work to find the three angles . (They can be found by, in principle, the cosine law on sphere. ) I would like to have a more direct formula using . So far the best (goodest?) result I can obtain is the following
Theorem 2 (Ken-Michael-Wongting’s theorem, 2011 (Area formula for triangle))
With the same notations as before,
Proof: By Theorem 1,
The plane has normal vector , and it is easy to see that the angle is the angle between and . Thus we have (see Lemma 2 of here)
Plugging these into (1), we can get the result.
Question: The formula looks quite complicated, can we simplify this result?