Ghost trails with Animate?

Question

What are some approaches to integrate Ghost trails methods with Animate, which otherwise seem more suited for exporting individual frames?

For example, given this rootPlot function, how can trails be added and displayed as a dynamic alterative to RootLocusPlot?

Animate[rootPlot[(Range[0, 20, 3] // Map[x^# &] // Apply[Plus]) + 
    k x^3, x] // Show[{Graphics@{Gray, Circle[{0, 0}, 1]}, #},
    PlotRange -> 2 {{-1, 1}, {-1, 1}}, Axes -> True] &,
 {k, -50, 50, 1}]

Answer 1

It really helps others reading your code if you split things up and name parts of the code. Let's start with the polynomial.

polynomial[k_] := Total[x^Range[0, 20, 3]] + k x^3;

and the function that finds the roots, and the function that visualizes them

roots[poly_, x_] := roots[poly, x] = Through@*{Re, Im} /@ (x /. NSolve[poly, x])
frame[k_, opacity_] := ListPlot[
  roots[polynomial[k], x],
  PlotStyle -> Opacity[opacity]
  ]

And the background can also have its own name so that it is clearly distinguishable in the code.

background = Graphics[{
    Gray, Circle[{0, 0}, 1]
    },
   PlotRange -> 2 {{-1, 1}, {-1, 1}},
   Axes -> True
   ];

Now, a ghost trail is nothing but a plot that includes the current frame and the last lag number of frames, with each frame faded by age.

visualize[k_, lag_, {min_, max_}] := 
 visualize[k, lag, {min, max}] = Module[{mink, frames},
   mink = Max[min, k - lag];
   frames = frame[#, (# - mink)/(k - mink)] & /@ Range[mink, k];
   Show[background, Sequence @@ frames]
   ]

And so finally:

Animate[
 visualize[k, 10, {-50,50}],
 {k, -50, 50}
 ]