Part 2 of the previous post on the Bezier Curve

Once root solutions to a have been found.  It seems the supplied parameter k then need be provided.  For point approximation on the curves interval we are only left with k since p need on only coincide with a relative maximal point on the curve approximation interval.  The simplest solution that I could think of (untested) at the moment should then rely on a numeric search algorithm which steps the interval of k from some arbitrarily selected value and then using known point comparison on the relative coordinate system.  Once having reached an adequate threshold for errors minimization on the curves point (between supplied points and those produced by the bezier curve), we can fix the value of k to such point.  The benefit of this method, as long as approximations aren't so arduous computationally speaking, is that certain points are thrown out defined by the third order smoothness provided by the bezier function.  Because of the potential number of solutions, however, and given the factor that k remains as of yet known in our search initially, not sure if this will be a great solutions approach.  Another, quick method for the generation of k involves a sample of two neighborhood points, containing the relative maximum and another, here generating an interpolated spline 3rd order equation from both sample points and subsequent first derivative (left right slopes).  Then from the generated spline, deriving k with a second order derivative on the interpolated function.  At least this may provide a faster approximation method at such point.