Geometry on the Cone

See how much geometry you can develop on cones with cone angle < 360 degrees (e.g., the 90 degree cone).  You may also be interested in comparing these with cones with cone angle > 360 degrees (e.g., the 450 degree cone) and/or cylinders.

For any of these surfaces you can ask any of the following questions:
What do the intrinsically straight lines look like?  What happens to lines that run into the cone point?
What do circles look like?
Which of Euclid’s Postulates are true?
What is the sum of the interior angles of a triangle?
What is the holonomy of a triangle?
Which triangle congruence theorems hold (or don’t hold)?
Is there a unique straight line joining any two points?  If not, for which points is there a unique straight line joining them?
Do any straight lines intersect themselves?  If so, how many times?

Explore any of these questions, or others, that interest you, and write a paper explaining what you find.