I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. another parametrization $\hat{\vec{r}}(t)=\langle \hat{x}(t),\hat{y}(t)\rangle$ that satisfies $\hat{x}'(t)^2+\hat{y}'(t)^2=1$ and $\hat{\vec{r}}\big((a,b)\big)=\vec{r}\big((a,b)\big)$.

What is the name of this theorem and who proved it? I've been looking around the internet and it seems like the Gauss-Bonnet Theorem comes up a lot, but I don't see the connection between that and this? Maybe I just don't understand it as well as I need to.

  • $\begingroup$ it is just a change of variables (the speed/derivative of $\vec{r}$ should be nonzero everywhere). Who might have invented that, I don't know $\endgroup$ – user66081 Jan 15 '14 at 19:33
  • $\begingroup$ I don't think you can have both conditions at once, unless $(a,b)$ on the left and $(a,b)$ on the right represent two different coordinate systems, which would be a strange way of writing. In any case the first condition is attained by taking $\hat {\vec r}= \vec r / \|\vec r\|$ $\endgroup$ – GPerez Jan 15 '14 at 19:42
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    $\begingroup$ It should be a curve where arc-length exists. Say $x(t), y(t)$ are $C^1$. $\endgroup$ – GEdgar Jan 15 '14 at 20:06

The earliest use is due to Leonard Euler in 1775: Methodus facilis omnia Symptomata linearum Curvarum non in eodem Plano sitarum Investigandi, Acta. Acad. Scient. Petropolitanse 1782 I, 1786, 19--57.


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