42 1. Various Ways of Representing Surfaces and Examples
Figure 1.24. Determining the area of a spherical triangle.
we have are the angles, however, we cannot determine the area, since
the triangle could be scaled up or down, preserving the angles while
changing the area.
This is not the case on the surface of the sphere; given a spherical
triangle, that is, the region on the sphere enclosed by three geodesics
(great circles), we can find the area of the triangle via a wonderfully
elegant formula in terms of the angles, as follows.
Consider the ‘wedge’ lying between two lines of longitude on the
surface of a sphere, with an angle α between them. The area of
this wedge is proportional to α, and since the surface area of the
sphere with radius R is
it follows that the area of the wedge
4πR2 = 2αR2. If we take this together with its mirror image
(upon reflection through the origin), which lies on the other side of
the sphere, runs between the same poles, and has the same area, then
the area of the ‘double wedge’ shown in Figure 1.24 is 4αR2.
Now consider a spherical triangle with angles α, β, and γ. Put
the vertex with angle α at the north pole, and consider the double
wedge lying between the two great circles which form the angle α.
Paint this double wedge red; as we saw above, it has area 4αR2.
Repeat this process with the angle β, painting the new double
wedge yellow, and with γ, painting that double wedge blue. Now
every point on the sphere has been painted exactly one colour (or,
as in Figure 1.24, one particular shade of gray), with the exception
of the points lying inside our triangle, and the points diametrically