Let me first state the original problem I want to solve:
Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$, and their coordinates $C(t_0),\cdots,C(t_n)$, find some interpolating curve $\tilde C$ such that their difference in length (denoted by $L$) satisfies: $$|L(C)-L(\tilde C)|\leq c_0h^4$$ where $c_0$ is some constant and $h$ is the maximum length of the intervals $[t_{i-1},t_i]$.
My idea is to use a cubic spline interpolation on $x$ and $y$ coordinates respectively. Then the difference in length can be computed by:
$$|L(C)-L(\tilde C)|\leq\int_a^b|C'(t)-\tilde C'(t)|dt$$
If we can prove $|C'(t)-\tilde C'(t)|\leq c_0 h^4$ then we are done.
The best error bound on cubic spline interpolation that I know of is from Hall and Meyer, Optimal error bounds for cubic spline interpolation, Journal of Approximation Theory, Vol. 16, 1976, pp.105-122. They show that if the interpolated function $f$ is in $C^4$, then the interpolating function $\pi f$ along with its derivatives has error bounds
$$\lVert f^{(r)}-(\pi f)^{(r)}\rVert_\infty\leq C_rh^{4-r}$$
There are some problems of applying this result to my original problem:
- The error bound for the first derivative is of order 3.
- Hall and Meyer proved their result for type-I and type-II boundary conditions, i.e. specifying either the first or the second derivatives at the boundary. However, since I am considering a closed curve, I actually need to prove an error bound for periodic boundary conditions.
On the other hand, since I have a $C^\infty$ instead of $C^4$, I also see a chance that the error bound can be improved. Any idea is greatly appreciated.
