How can I get a plot of "surface area coverage" of a parametric plot?

Question

I have a parametric function defined by the x and y functions:

x = r*Cos[w_0*t - w_L*t] + 
  z*Cos[w_P*t - w_L*t]

y = r*Sin[w_0*t - w_L*t] + 
  z*Sin[w_P*t - w_L*t]

I can change the variables around to see interesting patterns formed by the function. What I want to do is have some sort of plot of this 2-D function where a grid is placed over the plot, and each square of the grid has the total amount of coverage of the function over the grid, meaning the total length of all the the lines passing through it. From the image, you can see that grid 3 has much more coverage than grid 4. That's what I want to quantify, but I really don't know how to attack the problem.

Basically, I want to set my parametric plot flat on a table and see the variation of density in different regions. Thus grids with lots of coverage will have a higher z-value. Perhaps there are other solutions to this than my idea, but I am certainly not sure.

Answer 1

Here is the image-processing idea I hinted at in comments. Let's define some values and get a pleasing plot:

w0 = 15 Pi; wL = 21 Pi; wP = 35 Pi; r = 1.5; z = 15;
x = r*Cos[w0*t - wL*t] + z*Cos[wP*t - wL*t];
y = r*Sin[w0*t - wL*t] + z*Sin[wP*t - wL*t];

plot = ParametricPlot[
         {x, y}, {t, 0, 1},
         PlotStyle -> Black,
         PlotRange -> {0, Automatic},
         PlotRangePadding -> {{0, Scaled[0.05]}, {0, Scaled[0.05]}},
         Axes -> False
       ]

Then Rasterize the plot, Binarize the resulting image, then chop it up into 16 equal squares with ImagePartition:

(chopped = 
    plot // Rasterize[#, ImageResolution -> 150] & // Binarize // 
     ImagePartition[#, Scaled[1/4]] &) // Grid;

Calculate the density of coverage for each square using the mean intensity of the corresponding image:

(densities = 
   1 - ImageMeasurements[#, "MeanIntensity"] & /@ chopped // 
    Round[#, 0.01] &) // Grid

(* Out: 
0.13    0.08    0.01    0.
0.      0.06    0.12    0.01
0.      0.      0.06    0.08
0.      0.      0.      0.13
*)

I calculate (1-intensity) because that way high intensity/coverage corresponds to a low value, and viceversa; that helps with plotting later on.

Now combine the plot of the function with a visual representation of the density values, using ArrayPlot; note that plot range, data ranges etc need to be adjusted to match between the original plot and the density representation; I take the necessary values from the original plot:

Show[
  ArrayPlot[
    densities,
    DataRange -> PlotRange[plot],
    ColorFunction -> (Blend[{White, Red}, #] &)
  ],
  plot,
  PlotRange -> PlotRange[plot], 
  PlotRangePadding -> {{0, Scaled[0.05]}, {0, Scaled[0.05]}}
]