I've got a curve in $\mathbb R^2 : (x(s) , y(s)) $ ($s \in (0,L)$).
For the tangent, it holds: $t(s) = \left( \matrix{x'(s) \\ y'(s)} \right) $. Since we have arc length, it holds also $||t|| = 1$.
Let $\psi(s)$ s.t. $cos(\psi(s)) = x'(s)$ and $sin(\psi(s)) = y'(s)$.
The curvature is defined by $K(s) := \psi'(s)$
Let $f: (-R, R) \rightarrow \mathbb R, f(x) = \sqrt{R^2-x^2} $
Now (finally) the question: what's the parametrization of the graph of this function? (in arc length)