Thread: What is Straight on a Hyperbolic Plane?

a.) On a hyperbolic plane, consider the curves that run radially across each annular strip. Argue that these curves are intrinsically straight. Also, show that any two of them are asymptotic, in the sense that they converge towards each other but do not intersect.


b.) Find other geodesics on your physical hyperbolic surface. Use the properties of straightness (such as symmetries).


c.) What properties do you notice for geodesics on a hyperbolic plane? How are they the same as geodesics on the plane or spheres, and how are they different from geodesics on the plane and spheres?

By the "physical hyperbolic surface," do you mean tractricoid?

There is lack of context for your question. What have you covered about this subject? What does it mean to "argue" and what is "intrinsically straight"? Do you need to prove algebraically that the curves that run radially across each annular strip are geodesics? Then this questions is definitely not at the pre-university level.

I personally have not learned about these topics until my third year in college, so you want to ask these questions in the Topology and Differential Geometry forum. Otherwise, if you are studying this on an elementary level, you need to provide more information, such as what properties of straightness you covered.

Thank you for your help.

I do need to prove algebraically that the curves that run radially across each annular strip are geodesics?

I need a proof for this along with any information you have on this topic. The book I was given that teaches me about this topic is just a vauge as this question. I am not around my notes right now but I will post back and let you know. In the mean time, if anyone knows a proof of any kind this would be very helpful.

By the physical hyperbolic surface I do not mean tractricoid.

For part a we are looking for the local intrinsic symmetries of each annular strip and then global summetries in the whole hyperbolic plane. We need to make sure we give a convincing argument why the symmetry holds in the limit.

We shall say that two geodecis that converge in this way are asymptotic geodesics. Note that there are no geodesics (striaght line) on the plane that are asymptotic.

For part b we are holding two points between the index fingers and thumbs on our two hands. Now if we pull gently- a geodesics segment with its reflection symmetry should appear between the two points. If your surface is durable enough, try foliding he surface along a geodesic. Also, you may use a ribbon to test for geodesics.

For part c we are exploring properties of geodesics involving interesting, uniqueness, and symmetires. We are convincing ourselves as much as possible using a model- full proofs for some of the properties will have to wait.