1. Consider two distinct points p and q on each of the following surfaces. Is there always exactly one segment between them?
2. If not, what are the possibilities? Explain each case as completely as possible.Answer the question for the hyperbolic plane H²,
3. Answer the question for the sphere S². Given a third distinct point r, how many of the segments contain r?
4. Answer the question for the projective plane P².
5. The cylinder C.

Use the axioms to give a careful proof of your claims in parts a), b), c), and d).

These proofs should show why the answers are different.

Make sure that you consider each case and refer to the specific part(s) of the specific axiom(s) that you are using.