The problem (as often is in math) is finding $x$: \begin{equation} d = \int_{x_0}^x\sqrt{1+[f'(t)]^2} \ \text{dt} \end{equation} It's an inverse problem, and I'm looking for an approximation of the solution for computational purposes. I'm trying to distribute evenly arc-lenght spaced points along the graph of a given real function.
Using the Riemann sum for the integral gives me no hope...any help? EDIT: the Riemann-sum approach (with a subdivision of the interval $[x_0,x]$ in $N$ sub-intervals) looks like this, with $dt\simeq \Delta t:=\frac{1}{N}(x-x_0)$: \begin{equation} d = \Delta t\sum_{k=0}^N\sqrt{1+[f'(x_k)]^2}=\frac{x-x_0}{N}\sum_{k=0}^N\sqrt{1+\Delta t^2\left[f(x_0+(k+1)\Delta t)-f(x_0+k\Delta t)\right]^2} \end{equation} where we're using \begin{equation} f'(x_k)\simeq N\frac{f(x_{k+1})-f(x_k)}{x-x_0} \end{equation}