Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.research

Topic: geodesic question
Replies: 8   Last Post: Jul 30, 2011 2:30 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Lee Rudolph

Posts: 3,143
Registered: 12/3/04
Re: geodesic question
Posted: Jul 27, 2011 9:30 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

David Lowry <david.j.lowry@gmail.com> writes:

>On Jul 18, 2:43 pm, Mark Steinberger <nyj...@gmail.com> wrote:
>> Does anyone know an elementary proof that great circles in the sphere
>> minimize arc length?

>
>There is a triangle inequality in spherical geometry as well (as long
>as you consider points on the same half-sphere, which is always
>possible). If we are allowed to assume this, then it becomes very
>simple.
>
>Also, proving this is not so bad.


One ought also to prove that the metric for which the triangle
inequality has been assumed or proved (presumably the metric
such that the distance between P and Q is the product of the
radius of the sphere and the central angle between P and Q)
is identical to the metric generated by arc length (i.e.,
such that the distance between P and Q is the infimum of the
arc lengths of rectifiable curves joining P to Q), while
being careful not to make a (not great...) circular argument.

Taking such care, however, is likely to cause "elementary"-ness
to fly out the window. But since it's Mark's question, he
gets to decide on what "elementary" means in it. So far he
has not responded (publicly) to any of the proposed answers.

Lee Rudolph




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.