So there is a given set of control points $P_1, P_2... P_n$. I need to calculate additional 2 points for each pair of points, e.g. $S_{11}, S_{12}$ for $P_1$ and $P_2$.
From the 4 points $P_1, S_{11}, S_{12}, P_2$ I could create a cubic spline to smooth out the interpolation between the control points. So far I could only conclude that I need for all $i$ values $S_{(i-1)2}, P_i$ and $S_{i1}$ to be on the same line. But I have no idea how to calculate the velocities for each point.
So the question is, what algorithms are there to calculate the velocity (or directly these intermediate points) knowing only the control points?
There are so many question here regarding splines, but I did not find answer to this one. Sorry if this exists already. I only need help finding algorithms for that.
