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Prove the shortest path between two points is a straight line watch

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    The question asks me to show from first principles that the shortest continuous curve between two points is a straight line segment, in Euclidean space.

    How I would do it would be to say by the triangle inequality, going along one straight line segment is quicker than going along two straight line segments. Then blah blah blah, some analysis happens, and we see that every other path is longer than the straight line segment.

    This though isn't from first prinicples. I found a proof in this book Googlebooks on page 103. However, this uses things that I've never heard of (specifically between 9.8 and 9.9, stationarity). Is there another way to do it from 'first principles'?
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    It's really the same proof, but you might find an explanation you prefer if you google "calculus of variations" - which is the entire topic of finding functions that minimize integrals.
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    Mate, define it like that:p:

    But yeah, I thuoght you went to variational principles? He definitely proved it earlyish on there, although if you don't have the notes/don't have any idea whats going on in them, that would seem reasonable...
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    There is a problem with proving it using calculus of variations though - you end up only considering differentiable curves; and for us, it isn't first principles, since we don't use the integral definition of arclength. Gesar's proof should work for all continuous curves.
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    Zhen: good point - I read what I was expecting and not what was written.

    So what is your definition of arclength? (If I had to guess, something like the sup of the length of all polygonal 'samplings' of your curve. From which it's pretty trivial I think).

    Are you and Gesar doing the same course?
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    Slumpy: Didn't do Variational Principles. I went to the first lecture and slept through it.

    Our definition of curve length is pretty much exactly what you said, DFranklin. So I guess I'll bung in my original thoughts as an answer.

    And Zhen and I are doing the same course, yeah. Though I don't think I'll be doing it for much longer :tongue: Geometry doesn't learn me good, it seems.
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    (Original post by Gesar)
    Slumpy: Didn't do Variational Principles. I went to the first lecture and slept through it.

    Our definition of curve length is pretty much exactly what you said, DFranklin. So I guess I'll bung in my original thoughts as an answer.

    And Zhen and I are doing the same course, yeah. Though I don't think I'll be doing it for much longer :tongue: Geometry doesn't learn me good, it seems.
    Coulda sworn you were at more of it...but of course, you were just a met/top+optimization man..I remember you leaving in the middle now:p:
 
 
 
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Updated: February 6, 2010
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