What's a good algorithm for estimating the arc length of an algebraic curve?

Question

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.

Answer 1

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.