"Geodesics never intersect. (Though they may be finite in length because they "meet" themselves, eg a geodesic on a sphere is a circle.)"

I disagree. I am stuck on the intersecting part, and how many geodesics there are connecting two points.

For example, I know at 360 degrees there is only one.

At 180 degrees, there are 2

And at 90 degrees there are 3, but I have no way of proving this.

As far as intersections go, according to my professor, there are 2 intersections from 180 degrees to 90 degrees, 3 from 90-60 degrees..how do I prove this. He tried drawing these R^2 planes that was a digraph of R^3, however, it made no sense to me.

My professor agreed with me that there are any number of geodesics between any two points on a cylinder not on the same parallel. However, he disagreed when I said there is an infinite amount of geodesics between two points on a cone by simply choosing the degree of the cone (which I obtained from some online research). I got it from

Geodesics on a Cone which states, "So there can be any number of geodesics between two points on a cone, provided you choose θ appropriately." It makes sense to me after reading it.

I am really confused now.