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.  

Examination of the bezier curve under different conditions (followup from previous post)

I used http://live.sympy.org/ for symbolic computations at this point, another online site that nice provides math analysis tools.

I'd mention as in the previous posting on this subject matter, of supplying a rotation and translation of coordinate where it were assumed the component dx/dt on the parametric curve of the bezier function where neither simultaneously zero at a supplied point, and that we would assumed dy/dt were zero at such point regarding a time t  in the range of [0,1].

The supplied parametric equation on the rotated coordinate and translated coordinate frame for the relative y axis then can be described at such point of time t as


3at(−t+1)2+3bt2(−t+1)

equal to a point value p on the curves interval representing a relative y maximum.  Here points a and b are control points of the third order bezier function, noting that controls points y_0 and y_3 are simultaneously zero on the rotated coordinate axis.  Unlike the previous posting where assumptions were given such that a = b, we provide the possibility of control points a not equal to b in this examination.

The first derivative of this equation is

9at2−12at+3a−9bt2+6bt

noting at such a time t the  dy/dt = 0 by assumptions above.  
The second derivative of this equation is

18at−12a−18bt+6b

where at such time t we supply an additional parameter k equal to such equation.

Thus the second derivative equation at time t solved for t becomes eq.(1)

12a−6b+k18a−18b

                 
The first order equation at time t solved for b becomes eq.(2)

[a2−16−27a2+k2−−−−−−−−−√,a2+16−27a2+k2−−−−−−−−−√]

Assuming the the first equation in the equations set (2) for b implies with back substitution to eq.(1)

9a+k+−27a2+k2−−−−−−−−−√9a+3−27a2+k2−−−−−−−−−√

for time t providing eq.(3).

Thus we can see how large our original poly becomes with substitutions of eq.(3) and b from eq.(2)
into the parametric equation



1(3a+−27a2+k2−−−−−−−−−√)3(a9(−k+2−27a2+k2−−−−−−−−−√)2(9a+k+−27a2+k2−−−−−−−−−√)−p(3a+−27a2+k2−−−−−−−−−√)3+154(3a−−27a2+k2−−−−−−−−−√)(−k+2−27a2+k2−−−−−−−−−√)(9a+k+−27a2+k2−−−−−−−−−√)2)

here it appears a numeric method will be needed for roots solutions