I came across this paper here describing a method of creating an n_th order Bezier curve that approximates multiple other Bezier curves that are connected. The method is based on minimzing the error between the calculated curve from the original segments in the least squares sense. In the paper it was mentioned that there is a certain amount of error (since it is an approximation after all) but there was no quantification of this error.
My question is: are there ways to quantify this error in approximation? i.e. describing how well the fit will be with the chosen order of the fit Bezier curve for example?
Bezier curves are (parametric) polynomials, and there is around a century of numerical analysis research related to approximation of functions by polynomials.
Roughly speaking, the error depends on the smoothness of the functions you're trying to approximate. There are dozens of relevant theorems. A good place to start looking might be "Approximation Theory and Practice" by Trefethen. The first 6 chapters are freely available here.
One simple approach ...
Suppose your input is two curves $A_1$ and $A_2$, and you approximate them with a third curve $B$. Split $B$ at the point that's closest to the junction of $A_1$ and $A_2$. After splitting, $B$ has now become two curves $B_1$ and $B_2$. Now let's try to estimate the deviation between $A_1$ and $B_1$. The quantity $\max\{\|A_1(t) - B_1(t)\| : 0 \le t \le 1\}$ is bounded above by the maximum distance between corresponding control points of $A_1$ and $B_1$. The same sort of reasoning applies to $A_2$ and $B_2$.