1. This routine is intended for plotting 3D or 2D trajectories {x1(t), y1(t)}, {x2(t), y2(t)} with the following requirements:

(a) A pixel-by-pixel mode as though shooting with a machine gun so that the bullets (pixels) lie next to each other (or with the given pixel distance).

(b) If no real time mode was set, the plotting must take the shortest possible time, about a few milliseconds so that it be negligible in comparison with the user designated play time interval (usually many seconds). 

(c) If the real time mode is activated, each curve must evolve in a real time in accordance with time t and the length of the time interval designate for this process. Contrary to the fastest possible pixel plotting in the mode (b), the real time evolution of the curves should be achieved by adding a calculated delay after plotting each pixel so that summarily all the delays add up to the given time interval designated for playing these curves. 

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2. The curves are represented with their multiple Taylor expansions, each expansion representing a curvy segment. The routine should plot pixels comprising one of those curvy segments corresponding to the current node. The time length of a segment is denoted step.

3. "The conflict of interests". In order to meet the requirement (a), the routine must implement computation which accurately determines the proper time step dt such, that to avoid excessive pixels shot into the same pixel position, and yet that the shooting do not leave gaps in curves. To achieve this goal, ideally we would need a parametric representation of the trajectories depending not on t (as is given), but on the arc length s: {x1(s), y1(s)}, {x2(s), y2(s)}, … Then, given the desired pixel distance ds, we could merely increment s with the step ds obtaining the plotted pixels with the desired distance.

However the parametric representation depending on arc length is not available, and even if it were, it would not help to meet the requirement (b), because the plotting must evolve dynamically according to the real time t (rather then according to arc length s). That is why designed this algorithm as dt-driven rather than ds-driven.

4. The dt-driven algorithm utilizes the differential geometry formula

ds = sqrt( (x')2 + (y')2 )*dt  or  dt = ds / sqrt( (x')2 + (y')2 )  (*)

(x, y and all their derivatives are available, and ds is given). This formula allows to obtain the desired dt providing that the point is not stationary i.e. that sqrt( (x')2 + (y')2 ) > 0 . The graphs obtained according to this formula looks continuous and well if (x')2 + (y')is big enough, meaning

ds / sqrt( (x')2 + (y')2 ) = dt << step  or sqrt( (x')2 + (y')2 )>> ds / step

Thus, in vicinities of stationary points, this formula was not applicable, some heuristic assumptions were made, which could cause gaps in curves near such points.

5. In order to avoid gaps near stationary points, since the current version the algorithm was improved relying on the following procedures. Before beginning the plotting process for a given Taylor expansion within the time interval step, first the program computes an approximate curve length for the segment step for each of the trajectories approximating curves with the respective broken lines (consisting say of 10 straight segments). Let the longest of those lengths be L. Recalling about the given pixel distance ds, the value n=L/ds therefore stands for a number of pixels in L so that step/n = ds*step/L is the upper boundary for dt in order that curves have no gaps.

6. Therefore instead of computing dt according to formula (*), first program sets dt := L/ds . Then, if (x')2 + (y')2 > 0, the program modifies dt according to the improved formula

dt = min{ ds*step/L, ds / sqrt( (x')2 + (y')2 ) }

so that even in the vicinity of stationary points dt is not allowed to become big and cause any gaps.

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