I know that determining the length of a bezier curve is not trivial to say the least.
However, to try to prove the convergence of an algorithm, I would like to compare the length of a curve with respect to the same curve to which a waypoint has been added in a specific way. Formally:
Let $\mathbf{c}(t) = \sum_{i=0}^n B_i^n \mathbf{P}_i$ be a bezier curve of arbitrary dimension (at least 3).
Let $\mathbf{c}(t^*) = \mathbf{P}_{w}, t^* \in [0,1] $ be a point of $\mathbf{c}(t)$. We add $\mathbf{P}_{w}$ as a waypoint at position $j, 0 \leq j \leq n$ to define a new curve (Concretely, in general $j = n-3$ because I have a constraint on the end acceleration of $\mathbf{c}$):
$ \mathbf{c_w}(t) = \sum_{i=0}^{j} B_i^{n+1} \mathbf{P}_i + B_j^{n+1} \mathbf{P}_{w} + \sum_{i=j+1}^{n+1} B_{i+1}^{n+1} \mathbf{P}_i $.
My questions are thus:
Under what conditions is $\mathbf{c_w}(t)$ shorter or longer than $\mathbf{c}(t)$ ?
What can we say about the length of the derivatives of $\mathbf{c}(t)$ and $\mathbf{c_w}(t)$ ?
Thank you very much for your time.