I am writing a program where I need to generate the sequence of points $x_1$ , $x_2$, $x_3$,$...$ such that the corresponding ($x_1$, $y_1$), ($x_2$, $y_2$), ($x_3$, $y_3$), $...$ on a generic function $y = f(x)$ are as close as we like but always equispaced. The distance is measured as length on the curve $f(x)$ [you might imagine, for instance, taking a semicircle and subdividing it into many small equal arcs, the widths of the slices will be different].
I can assume to restrict to an interval of the $x$ axis and make the subdivisions arbitrarily numerous and small.
Is there some exact asymptotic expression for the generic $x_i$ of the sequence?
(does this problem have a name in math?)
I think that, if we look for a generic solution, it would be necessary to use a local approximation of the function $f(x)$ to obtain an approximation of $$L=\int_a^b \sqrt{1+[f'(x)]^2}\,dx$$
Using a Taylor expansion of $f(x)$ around $x=a$
$$L= (b-a) \sqrt{f'(a)^2+1}+(b-a)^2\frac{ f'(a) f''(a)}{2 \sqrt{f'(a)^2+1}}+(b-a)^3\frac{f''(a)^2+f'''(a) f'(a) \left(f'(a)^2+1\right)}{6 \left(f'(a)^2+1\right)^{3/2}}+O\left((b-a)^4\right)$$ Then, using power series reversion, $$b=a+\frac{L}{\sqrt{f'(a)^2+1}}-\frac{L^2 \left(f'(a) f''(a)\right)}{2 \left(f'(a)^2+1\right)^2}+$$ $$\frac{L^3 \left(\left(3 f'(a)^2-1\right) f''(a)^2-f'''(a) f'(a) \left(f'(a)^2+1\right)\right)}{6 \left(f'(a)^2+1\right)^{7/2}}+O\left(L^4\right)$$ we could take as many terms as we wish