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Suppose $\gamma(t)$ for each $t$ is a curve. We may write $$\gamma(t)(s) = \gamma_0(s) + d(t,s)N(s)$$ where $\gamma_0$ is some fixed curve, $N$ is the unit normal vector and $d$ is a distance from $\gamma_0$ to the curve $\gamma(t)$.

Under what conditions can this be done? I vaguely recall we need $\gamma_0$ to be Lipschitz, but don't know why or how. And for what $t$ can be this done? Appreciate any references or explanation.

Thanks

Pink Panther
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  • There is not a clear question here. "We may write ..." Of course you can always write that. You've written a definition down, so I don't see a reason for there to be a problem unless there are some more conditions going on (also, what is $N(s)$?). – Matt Aug 13 '12 at 15:49
  • @Matt Sorry, $N$ is the unit normal vector. My question is, given $\gamma(t)$ and $\gamma_0$, what are the conditions under which that equation I wrote down makes sense? That may be unclear but I don't have any more details I'm afraid. I just heard about it and want to know more. – Pink Panther Aug 13 '12 at 15:56
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    One can always make a parallel curve from a given curve... – J. M. ain't a mathematician Aug 13 '12 at 16:03

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