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Is a straight line the shortest path?

Published online by Cambridge University Press:  08 February 2018

Jessica E. Banks
Jessica E. Banks*
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW e-mails: jessica.banks@lmh.oxon.org, jessica.banks@bristol.a.c.uk
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Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side.

The same sorts of considerations apply in mathematical worlds. I use the mountainside image because it is my preferred way of thinking of a Riemannian metric. Pick an abstract surface S. A Riemannian metric on S gives a well-behaved distance function. By force of habit I tend to picture S as sitting somehow within the physical world. Probably, I'm looking at it from the outside. But if I change viewpoint, so that I am walking around on S, I can picture how the topography affects the idea of the ‘shortest path’.

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 553 , March 2018 , pp. 1 - 12
Copyright
Copyright © Mathematical Association 2018 

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