Section 8.6 Lunes
¶A lune is the region formed between two great circles on a sphere, as illustrated in Figure 8.6.1. 1 Analogously, a lune is the region between two elliptic lines in the Klein disk, as illustrated in Figure 8.6.2. 2
Activity 8.6.1.
What does a Klein lune look like if its vertex isn't on the equator?
Since we know that the surface area of a sphere is \(4\pi r^2\text{,}\) the area of a lune with angle \(\alpha\) is obtained by proportional reasoning as
Since the Klein disk can be thought of as the Northern Hemisphere, its area must be \(2\pi r^2\text{,}\) but, due to the wraparound feature, the entire disk is covered when \(\alpha=\pi\text{.}\) Thus, the area of a Klein lune is given by
which is the same as the area of the corresponding spherical lune.
Imagine now an elliptic triangle. Construct a lune at each vertex using the angles of the triangle. On the Klein disk, these three lunes already cover the entire disk, as shown in Figure 8.6.4. And more, as the triangle itself is contained in each lune. Letting \(T\) denote the area of the triangle, we therefore have
or equivalently
where \(S\) denotes the angle sum of the triangle. We have therefore shown that the area of a triangle in the Klein disk is a constant (\(2r^2\)) times the angle excess of the triangle.
The same argument works on a sphere. However, lunes on a sphere always come in pairs, since the great circles continue past their intersection points. The construction above therefore yields six lunes, but also six copies of the triangle. The computation now reads
which yields the same conclusion