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Curvature Circle Proof

Here is the problem:

Show that if [tex]c[/tex] is a curve with [tex]\kappa=\frac{1}{r}[/tex] (r is a positive constant) that [tex]c[/tex] is moving on a circle of radius r.

He gives a hunt to use the formula [tex]E(s)=C(s)+rN(s)[/tex]. I don't know where he got this equations and I have no idea what the function E is supposed to represent. I'm sure C and S are position and arclength respectively. So first I showed that [tex]\frac{dE}{ds}=0[/tex] with the definitions of T and N vectors as related to curvature K.

Then he gives a hint to show [tex]absolute value(C-E)=r[/tex] which I have no idea how to show, and then from that to explain why that makes C a circle or radius r?

I know that the equation for a circle is nx^2+ny^2=r^2 but I don't see where that will get me here.

Any help?

I know this isn't in the correct format but this is more of a rigorous proof than a problem with given information...
 
Last edited:

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