Geometric warm-up

Problem 1: triangle on a sphere

What is maximum sum of angles in a triangle on a sphere?

solution

$540^\circ$.

Problem 2: circle on a sphere

$^*$ Consider the sphere of radius $R$. A circle is drawn on the sphere which has radius $r$ as measured along the sphere. Find the circumference of the circle as a function of $r$.

solution

$\displaystyle L = 2\pi R\sin {r \over R}$.

Problem 3: density on a sphere

Suppose that galaxies are distributed evenly on a two-dimensional sphere of radius $R$ with number density $n$ per unit area. Determine the total number $N$ of galaxies inside a radius $r$. Do you see more or fewer galaxies out to the same radius, compared to the flat case?

solution

$N = 2\pi nR^2 \left( {1 - \cos {r \over R}} \right) \approx n\pi R^2 \left( {1 - {{r^2 } \over {12R^2 }}} \right) < n\pi R^2 $.

Problem 4: angular sizes in spaces of constant curvature

An object of size $A$ is situated at distance $B$. Determine the angle at which the object is viewed in flat space and in spaces of constant (positive and negative) curvature.

solution

In a flat space we have \[\alpha = \arccos \left( {1 - \frac{A^2}{2B^2}} \right),\] while in a space of constant negative curvature with radius of curvature $R$ \[\alpha = \arccos \left( 1 - \frac{\operatorname{ch} \left( {A/R} \right) - 1}{\operatorname{sh} ^2 (B/R)} \right).\] When $R \to \infty $ this case turns into the flat one. The expression for constant positive curvate could be obtained by replacement $R \to iR$.