I have a question regarding the re-parameterisation of a B-spline.
Some info:
- The B-spline is of order 4 (degree 5), hence $C^3$ continuity
- There is no knot multiplicity
- The end conditions are not relevant/important
- The goal is to interpolate (or approximate with a given small threshold) a set of data points
- There must not be any loops or large bulbs after re-parameterization
- The derivatives need to be continuous after re-parameterization
What would be the best method to have as low as possible $C^2$ fluctuations? I have read about centripetal parameterization.
Secondly which book describes these parameterisation methods best?
Lastly: I need some better understanding of the following. Assume that a point mass is travelling along the spline at a given velocity. How does the derivative affect the mass's acceleration and jerk taken into account that the derivatives are with respect to the spline's parameter, e.g. not the distance along the spline itself (chord length). Do I need to re-parameterize somehow to have continuity with respect to the distance along the curve? This aspect seems either not discussed very often or is not relevant?
First, I think your first bullet has a typo or two. If your curve has single knots and is $C_3$, then it probably has degree 4 (order 5).
Assigning parameter values to data points (the points to be interpolated) is not really a science, it's more a set of heuristics. Chord-length and centripedal parameter values usually give good results, but they are certainly not optimal (whatever that means). They are used because they're easy to compute. If you have some "curve quality" function that you want to optimize, then you really should treat the parameter values as independent variables in the optimization process. Unfortunately, this makes the optimization much more difficult.
A good place to start reading about parameterization is Farin's book. But, you won't find much beyond what I wrote above.
To understand velocity, acceleration, forces, you need to know the position of the moving particle as a function of time. If you know that the speed is constant, then you can get acceleration using standard formulas from differential geometry. If speed is not constant, then you need some information about position or speed or distance travelled as a function of time. The shape of the curve does not provide enough information, by itself.