Is the Geodesic Always the Shortest Path Between Two Points?

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Discussion Overview

The discussion revolves around the nature of geodesics and whether they always represent the shortest path between two points on various surfaces, including examples such as cylinders and toruses. The scope includes theoretical considerations and examples from differential geometry.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants assert that geodesics are defined as the shortest paths between two points on a surface, raising the question of whether this holds true universally.
  • Others argue that on certain surfaces, such as cylinders and flat toruses, there are infinitely many geodesics between most points, suggesting that not all geodesics are the shortest paths.
  • A participant introduces an example of a geodesic on a fluted surface of negative curvature that demonstrates a complex path, contrasting it with the shortest geodesic between adjacent points, which is a simple arc.
  • Another participant draws a parallel between geodesics and magnetic lines of force in a tokamak, noting their helical nature.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the relationship between geodesics and shortest paths, particularly in the context of different surfaces.

Contextual Notes

Limitations include the dependence on the specific geometry of surfaces and the potential for multiple geodesics existing between points, which complicates the assertion that geodesics are always the shortest paths.

enricfemi
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it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
then must the geodesic connected two points be the shortest path?
if not, what about the example?
Thanks for any reply!
 
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enricfemi said:
it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
then must the geodesic connected two points be the shortest path?
if not, what about the example?
Thanks for any reply!

check out the example in wikipedia
http://en.wikipedia.org/wiki/Geodesic
 
enricfemi said:
it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
then must the geodesic connected two points be the shortest path?
if not, what about the example?
Thanks for any reply!

No. On a cylinder there are infinitely many geodesics between most points. The same is true of a flat torus.
 
wofsy said:
No. On a cylinder there are infinitely many geodesics between most points. The same is true of a flat torus.

yeah, cylinder is really a good example!
 
There is an example of a geodesic on a fluted surface of negative curvature that winds almost all of the way down the surface circling around it in a helical motion then turns around and comes back! The shortest geodesic though between two adjacent points is a simple arc. I will try to look this up. It is pretty incredible.
 
it reminds me the magnetic lines of force in tokamak. they are all helical.
 

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