We know that the path of a point on a Bezier curve can be controlled through the control points of the curve. This parametric curve ranges from t=0 to t=1. This means the velocity is uniform for fixed control points. Is there any research where the possibilities of a non uniform (assigned) velocity for a point on a Bezier curve has been discussed?
The velocity in not uniform for fixed CP, you probably mean that the velocity will not change as long as the CP are not moved.
You can compute the direction of the velocity which is given by the slope of the bspline curve dy/dx [(dy/du)/(dx/du)] which is given in the NURBS book. As for the magnitude of the velocity it is for you to specify or to solve for based on the equations governing the phenomenon that you are trying to study.
Wish you all the best.
For your Bezier curve problem with non-uniform velocity, you may want to refer to the following paper:
Hwang, Jung-Hoon, Ronald C. Arkin, and Dong-Soo Kwon. "Mobile robots at your fingertip: Bezier curve on-line trajectory generation for supervisory control." Intelligent Robots and Systems, 2003.(IROS 2003). Proceedings. 2003 IEEE/RSJ International Conference on. Vol. 2. IEEE, 2003.
You have x(t), y(t) a derivatives etc. You need to reparametrize parametert t, is. you need to find a function t=g(v) so that the substitution that leads to the requred behavour of the Bezier curve. It means that you will get x(g(v)) and y(g(v)) .
Might not be quite easy way.
Vaclav
As Vaclav mentioned you need to have a reparameterization of the path, either analytically or using finite differences. There is a lot of information on the web, e.g. you may start with looking at "Keyframing by Interpolation" on
http://caig.cs.nctu.edu.tw/course/CA/
Henning.
Assuming that you are generating the Bezier curve using the de Casteljau algorithm on the interval [0,1], you can apply the square root on the parameter before applying the de Casteljau algorithm.
Good points above. In practice what you need is to reparametrize by arc length. Rick Parent's animation book covers this nicely. A numerical approach is to build a table of arc lengths using a linear approximation. Small steps in the parameter 't' give points on the curve. Calculate the linear distance between each pair and enter the total distance travelled in the table as (Euclidean) distance against parametric distance. This can be normalized by dividing by the total distance to give a 0
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