Proving the Relationship Between Chord Length and Curve Type | Homework Question

[akoska]

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.

 

[Dick]

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.

 

[akoska]

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

 

[Dick]

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).

 

[akoska]

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?

 

[Dick]

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.