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Prove the shortest distance between two points is a line

  1. Feb 16, 2017 #1
    1. The problem statement, all variables and given/known data

    Let γ : [0, L] → Rn be arclength parametrized. Show that the distance between the endpoints of the curve can at most be L, and equality can only hold when γ is a straight line segment. Thus, the shortest path between two points is the straight line segment connecting them.

    2. Relevant equations

    I guess maybe arclength: s(t) = ∫ ||γ'(t')||dt' from 0 to t

    3. The attempt at a solution
    So my attempt would be to include the properties of an arclength parameterized curve; Namely that the length of such a curve is 1. Such that when you compute the arc length you get something like ∫1dt' from 0 to t (or in this specific case 0 to L) this tells us the length is L but I don't know how to prove that the equality holds only when its a straight line. Any suggestions?
     
  2. jcsd
  3. Feb 17, 2017 #2

    pasmith

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    *cough* Calculus of variations *cough*
     
  4. Feb 17, 2017 #3

    FactChecker

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