I am not an expert in differential geometry; I hope the folks here can help me with this question. We have a 1D curve-line projected on a 2D Space. The curve is defined as follows: "y=f_curve(x)". We also know the domain and range of this curve. We are interested to "drop" equidistant points on this curve. The (arc-length) separation between the points is given by "s0".
Question --> How can we find the respective (x,y)-coordinates of each of these equidistant points on the curvy line? i.e. is there a generic mathematical approach to solve for the coordinates that can be applied to any surjective line-curve on a 2D plane? Of course these points can be found using brute-force technique; but is there an optimal/sub-optimal way to describe/solve this?
Animated image that visually describes this geometrical problem. Thanks!
Best Regards, M.A.
If by some chance the arclength integral can be found in elementary terms, then applying Newton's method to find roots of $$ arclength(s) = k s_0, k = 0, 1, \ldots, n $$ works pretty well. If not...then you're going to have to numerically approximate $arclength(s)$, and to do that accurately, you're going to need to know the kinds of things that @AndrewDHwang asked in the comments. For instance, if the "bumps" shown in your picture have width about $1$, then a numerical estimate using steps (in $x$) of size $\frac{1}{20}$ should let you get a decent-looking result; steps of size $\frac{1}{2}$ will probably lead to nonsense. Steps of size $10$ will generate noise rather than useful information. :(
One sad thing is that error accumulates: if your estimate of the first break-point is a little wrong, then your estimate of the next will include that error, etc. And in general, the arclength estimated by summing up lengths of straight-line segments is an underestimate of the true arclength, because straight lines are the short way to get from here to there. So you'll be accumulating errors that are all of the same sign, and that's unfortunate.
You could try approximating your curve with circle-arcs, and summing the lengths of those, but that's only a good solution if (a) your step size is so small that the circle-arc has angle measure less than about $\pi/10$, and (b) you have something that'll tell you the derivative of $f$ as well as $f$ itself.
From your response to Andrew's question, it looks as if this is a practical rather than a theoretical question.
One thought: it's possible that your function $f$ comes from measurements rather than being exact. If so, then there's an implicit error in the arclength, and you should probably not be trying to make "steps" that are anywhere near, in size, to this "measurement error".
But as Andrew says, the first thing you really ought to determine is bounds on the first two derivatives, if that's possible.