Search All of the Math Forum:

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

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Lee Rudolph Posts: 3,143 Registered: 12/3/04
Re: geodesic question
Posted: Jul 27, 2011 9:30 PM

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

>On Jul 18, 2:43Â 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

Date Subject Author
7/18/11 Mark Steinberger
7/19/11 mjc
7/21/11 Lee Rudolph
7/21/11 Ilya Zakharevich
7/21/11 WimC
7/21/11 Arthur
7/30/11 Celeste Deligne
7/27/11 David Lowry
7/27/11 Lee Rudolph