Existence of regular spherical n-gons

Question

I'm having problems with the solution of the following exercise.

Let n $ \geq 3$.

a)What is the interior angle of a regular Euclidean n-gon? Given the side length, what is its area?

b) Show that for any $\alpha_n$ $\lt$ $\beta$ $\lt$ $\pi$ there exists a regular spherical n-gon with interior angles $\beta$. What is its side length and area?

I have already solved a) but find it difficult to solve b). For proving the existence it might be sufficient to take n planes passing through the orign intersecting the sphere while also intersecting each other such that each plane intersects two others in such a manner that the angle contained between them is $\beta$. This should generate a family of great circles intersecting each other such that the desired n-gon is generated.

As per the construction above each side is a line segment of a great circle and hence the length of a side is given by the value of the spherical metric d(p,q)=arc cos $\langle\ p,q\rangle$ where p,q are two vertices of the n-gon that are connected by a side.

The area can be computed in a manner similar to the computation of the area of a euclidean n-gon by breaking up the spherical n-gon into n isoceles spherical triangles with interior angles $\frac{\beta}{2}$,$\frac{\beta}{2}$, $\frac{2\pi}{n}$ calculating the area of one triangle and then summing all the areas up.

I have two problems with this solution:

1) How can I show that a family of planes used for my construction of the n-gon does exist?

2) How can I compute d(p,q) in terms of $\beta$?

Answer 1

Choose a ray emanating from the center of the sphere, and an arbitrary point $P$ on this ray. In the plane perpendicular to the ray and passing through $P$ draw a regular $n$-gon having $P$ as its center and side lengths of $1$. Each edge of this $n$-gon, along with the center of the sphere, forms a plane that intersects the sphere in a great circle, and the triangle formed by the center and $n$-gon edge contains an arc of the great circle. The $n$ arcs found this way form a regular spherical $n$-gon.

By moving $P$ in on the ray, the spherical $n$-gon thus formed will grow larger, with everything changing continuously, until the interior angles reach $\pi$ as $P$ reaches the circle center. By moving $P$ out on the ray, the spherical $n$-gon will grow tiny. But in doing so, the curvature of the sphere will grow comparatively greater, resulting in the spherical $n$-gon becoming closer and closer to a planar $n$-gon, whose interior angles are $\alpha_n$.

Since this process is continuous, every interior angle between $\alpha_n$ and $\pi$ must be passed through. I'll leave the analytic side to you. You can use geometry to determine the angle between two planes (and therefore between two arcs) when $P$ is a particular distance $d$ from the circle center. This can be used to determine the relationship between $\beta$ and the arclength of a side of the spherical $n$-gon.