I would like to calculate the arc length of an already-interpolated piecewise cubic spline, where each segment is defined by a normal cubic polynomial ax^3 + bx^2 + cx + d. I am not sure, however, what the best route to take is.
My first idea is to use numerical integration and the following arc length formula to calculate the arc length for each segment and then sum them up: https://tutorial.math.lamar.edu/classes/calcii/arclength.aspx
I am not sure if this is the best approach, as I have minimal experience in numeric integration. If this is the approach to take, which numeric integration method should I use? If not, how can I accomplish this?
Thanks a lot
There is a closed-form expression in terms of the Elliptic integrals, but the exact computation is better done by Mathematica, and next you will need the elliptic functions handy.
The numerical method by polyline approximation (as in the link) is a little too elementary. For such a smooth function, Simpson's rule will be fine. https://en.wikipedia.org/wiki/Simpson%27s_rule