Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s \geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I want to fit this $c(s)$ curve (say, a quadratic polynomial), so that it smoothly merges with $t(s)$. Basically, as $s$ increases, $c(s)$ and $t(s)$ should end up being pretty much the same. For simplicity, let's say that both $t(s)$ and $c(s)$ are quadratic curves.
I am guessing I need to optimise some kind of objective involving both the distance between $c(s)$ and $t(s)$, as well as the derivative of $c(s)$. Of course, there will be a hyperparameter to control how smooth we want to be, vs. how fast we want the merging to happen.
The figure below shows what I mean by "smooth merging".
Is there any established method to deal with this problem?
I also need this to be computationally fairly fast, but that shouldn't be a problem for a 1D problem.