I have a cubic Hermite spline in three dimensions defined by $$\mathbf{p}(t)=(2t^3-3t^2+1)\,\mathbf{p}_0+(t^3-2t^2+t)\,\mathbf{m}_0+(-2t^3+3t^2)\,\mathbf{p}_1+(t^3-t^2)\,\mathbf{m}_1,$$ where $\mathbf{p}_0$, $\mathbf{m}_0$, $\mathbf{p}_1$, and $\mathbf{m}_1$ are arbitrary vectors and $t \in [0, 1]$.
I'd like to numerically evaluate the total arc length over $t$, i.e. $$\Delta s = \int_0^1 \sqrt{\mathbf{p}^\prime(t) \cdot \mathbf{p}^\prime(t)} \,\text{d}t.$$ I think it would be most efficient to use standard Gaussian quadrature, but I'm unsure what order of approximation to use. Since the integrand above is not a polynomial, I know I cannot choose an exact approximation, but can I in general limit the error to say $\leq\!1\%$?
Since the integrand is the square root of a fourth-order polynomial, I wonder if it can be satisfactorily approximated as a second-order polynomial. I can in principle evaluate the Taylor series of the integrand, but I don't know how to decide at what point the terms become negligibly small in general.
$$\int_0^1 \sqrt{P(t)}dt$$
where $P$ is in general a fourth degree polynomial.
You have therefore a closed form formula in terms of elliptice functions.
See here.
– Jean Marie May 13 '21 at 14:20