# Geodesic and the shortest path

1. Feb 18, 2010

### enricfemi

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!

2. Feb 19, 2010

### physixlover

check out the example in wikipedia
http://en.wikipedia.org/wiki/Geodesic

3. Feb 19, 2010

### wofsy

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

4. Feb 19, 2010

### enricfemi

yeah, cylinder is really a good example!

5. Feb 20, 2010

### wofsy

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.

6. Feb 21, 2010

### enricfemi

it reminds me the magnetic lines of force in tokamak. they are all helical.

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