# Curvature Circle Proof

by americanforest
Tags: circle, curvature, proof
 P: 218 Here is the problem: Show that if $$c$$ is a curve with $$\kappa=\frac{1}{r}$$ (r is a positive constant) that $$c$$ is moving on a circle of radius r. He gives a hunt to use the formula $$E(s)=C(s)+rN(s)$$. 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 $$\frac{dE}{ds}=0$$ with the definitions of T and N vectors as related to curvature K. Then he gives a hint to show $$absolute value(C-E)=r$$ 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...

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