spherical light

Homework

1. Prove that two great circles intersect.
2. Prove that two great circles bisect each other.
3. The sphere of radius one can be considered as
   
   In these two coordinate systems determine the distance along great circles between two arbitrary points on the sphere as a function of the coordinates.
4. A pole of a great circle is one of the endpoints of a diameter of the sphere perpendicular to the plane of the great circle. For example, the North and South poles are both poles for the equator. Prove that through a given point, not on a given great circle and not the pole of the great circle, there is a unique great circle ghrough the given point perpendicular to the given great circle.