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Plane curves question

  • Thread starter akoska
  • Start date
  • #1
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Homework Statement



I want to show that if the chhord length ||f(s)-f(t)|| depends only on |s-t| then the f is part of a line or circlle. f may not be regular or unit speed.

Homework Equations





The Attempt at a Solution



I'm trying to differentiete it and taking ||f'(t)||, but I really need help, because it's not working for me.
 

Answers and Replies

  • #2
Dick
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You should be able to show |f'(t)| is a constant. For the rest of the problem I would think about trying to express the curvature of f(t) in terms of things like |f(s)-f(t)| as s->t and hence argue that it is also a constant.
 
  • #3
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How do I show |f'(t)| is a constant? I get to the part: |f'(t)| = lim(dt->0) |a(dt)/dt| where a is the function ||f(s)-f(t)|| =a(|s-t|)... and then i'm stuck
 
  • #4
Dick
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Divide both sides of your last expression by |s-t| and let s->t. I would conclude |f'(t)|=a'(0) for an appropriately defined a. Or this "|f'(t)| = lim(dt->0) |a(dt)/dt|". That looks to me like the definition of a'(0).
 
Last edited:
  • #5
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Yes, but I think the 'appropriately defined a' is the catch. a can be any differentiable function, adn although i can show that |f'(t)| is a constant for certain a, how can I show it for all a?
 
  • #6
Dick
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There's no 'catch'. You've defined a(|s-t|)=|f(s)-f(t)|. That defines a(x) for non-negative x. The only 'appropriately defined' case is whether you bother to define a(x) for negative x or keep referring to one sided derivatives. It's really nothing.
 
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