# Curvature Circle Proof

1. Jan 17, 2007

### americanforest

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...

Last edited: Jan 17, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?