91Ó°ÊÓ

In Exercises 3.1-3.10, consider the geometry of the sphere \(S^{2} \subset \mathbb{R}^{3}\) of radius 1 with the intrinsic (spherical) metric. (a) Define, by analogy with Euclidean geometry, the notions of spherical circle and spherical disc with centre \(P \in S^{2}\) and radius \(\rho\). (b) Prove that a spherical circle with radius \(\rho<\pi\) has circumference \(2 \pi \sin \rho\). (c) Prove that a spherical disc of radius \(\rho<\pi\) has area \(2 \pi(1-\cos \rho)\).

Show that in polar coordinates $$ x=r \cos \theta, \quad y=r \sin \theta, \quad z=\sqrt{1-r^{2}} $$ on the sphere \(S^{2}\) of unit radius, the element of area in \(S^{2}\) is $$ \mathrm{d} A=\frac{r \mathrm{~d} r \mathrm{~d} \theta}{\sqrt{1-r^{2}}} $$ [Hint: consider a small sector \([\theta, \theta+\delta \theta] \times[r, r+\delta r]\) in \(\mathbb{R}^{2}\). Prove that the sector of \(S^{2}\) lying over it is very close to a spherical rectangle with length of sides equal to \(r \delta \theta\) and \(\left.\delta r / \sqrt{1-r^{2}} .\right]\)

Here is a general project: take any result you know in plane Euclidean geometry, find an analogue for spherical geometry, and either prove or disprove it. As concrete exercises, prove or deny the following: (a) the 3 medians of a triangle intersect in a point \(G\); (b) the 3 perpendicular bisectors of a triangle intersect in a point \(O\); (c) (harder) the 3 heights of a triangle intersect in a point \(H\).