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There are many very useful computational techniques for calculating the arc length of parametric curves. See for instance this paper and references therein. What I could not find easily was a similar study for non-parametric ones. Specifically, cubic algebraic curves.

My question is: can I do better than the chord length algorithm where I estimate the arc length by summing the individual line segments connecting a sample of points on the curve?

I strongly believe there must be a better (faster converging) method, especially since for every sampled point on the curve, I can easily compute the normal vector and (not so easily) other geometrical information, such as curvature, etc.

Edit: The curves I'm working with have a single component in the region of interest, which is a well defined simplex. Also, for these curves I have managed to develop an adaptive sampling technique which I use to find any number of points up to any precision.

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Can you even with confidence find any points on an algebraic curve? Even determining the number of connected components can be a mess. Consider something like

$$ (\frac{x}{1000})^{2} + (\frac{y}{1000})^{2} + A = 0 $$

For $A > 0$, the curve is empty; for $A$ just slightly less than $0$, the curve is a large-diameter circle. This shows that the arclength is ill-behaved as a function of the coefficients .... so unless you have great numerical precision, you're likely to miscalculate it.

You might say "It's OK...I'll just work with infinite-precision rationals" but are you certain that your curve contains any rational points?

TL;DR: I suspect this is hopeless to try to do well in general.

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  • $\begingroup$ Thanks for your answer. I have already sorted out the potential issues you describe. Please see updated question. $\endgroup$ Commented Jan 9, 2017 at 14:09

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