1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Homework Helper

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


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted