Angle variation for equally distant points on a sphere?

Question

Imagine seeing a sphere from a distance, upon it you put a ruler in which you note 60 (example) points on the sphere that are equally separated on the ruler.

How can one find the angles ($\alpha_1 , \alpha_2, \alpha_3, ...$) between the center of the sphere and you (the observer) for each of those points?

Answer 1

Let $\beta_n = \alpha_1 + \ldots + \alpha_n$. Then $r\sin\beta_n = cn$, where $c$ is the distance between adjacent points on the ruler, and $r$ is the radius of the sphere.

So $\alpha_n = \beta_n - \beta_{n-1} = \sin^{-1}(cn/r) - \sin^{-1}(c(n-1)/r)$.