This post is about the spherical cosine law which perhaps is well-known to many of you. However I am putting it here because I love and only know very simple geometry.
Let be a triangle on the unit sphere , i.e. the 3 sides of consists of geodesics segments on the sphere. Let be the three sides opposing the three angles respectively (we denote the vertices and the angle at these vertices by the same symbols).
Then we have the cosine law for spherical triangle
When I first arrived at this formula, I did not realize that this is actually the spherical cosine law, which is closely related to the Euclidean cosine law for triangle:
Actually the above cosine law can be viewed as the limiting case of the spherical version, and in turn the spherical cosine law is derived from Euclidean geometry. We will look at how these two are related after the proof.
The proof is quite simple. We can without loss of generality assume that is the north pole and lies on the plane. Note that then lies on the longitude whose plane containing it makes an angle with the plane (why?). Since the radius of the sphere is 1, the angle made with the z-axis is exactly the length of (arc-length), which is by definition, and similarly the angle made with the -axis is . So in spherical coordinates, the 3 points are given by
Now the distance , which is c by definition, is the same as the angle between and by the previous remark. So by applying dot product to the vectors and , we arrived at
This finishes the proof of (1).
We now show that (2) can be viewed as the limiting case of (1). Intuitively, the spherical triangle becomes flatter and flatter when the sides a, b tend to zero, and thus more and more like a Euclidean triangle on a plane. To get (2), note that the Taylor series of and are given by
Applying these to (1) except the term , and dropping all the term of order higher than 2, we have
Thus the Euclidean cosine law (2) is the limiting version of the spherical cosine law as the sides tend to zero by dropping the third or higher order terms (which corresponds to the “curvedness” of the sphere).