Creating Transparencies
by Harry Calkins and Theresa Fulton

rcoord := {Random[Real, {-1, 1}], Random[Real, {-1, 1}], 
        Random[Real, {-1, 1}]}

pts = Table[Point[rcoord], {80}];

plot1 = 
        Show[Graphics3D[{PointSize[ .02], pts}, PlotRange -> All, 
            SphericalRegion -> True, Boxed -> False, 
            AspectRatio -> 1, ViewPoint -> {2,  0, 0}, 
            Axes -> False]];

This pulls out the coordinates of each point.

pts = 
        Map[Last[Last[#]] &, Graphics[plot1][[1]]];

This assigns a radius to the coordinates so that you are forming circles.  This is important when you use the function that shades the inner region of the overlapping circles.

newpts = List[MapThread[List[#] &, {pts}, 1]];
mypts = MapThread[Append[#, 0.2] &, newpts, 1];

We are now pulling out the first coordinates of the two circles that will overlap.

mypts[[1, All]]

mypts[[2, All]]

Next, we will be using the function provided by my colleague.  His function uses the Graphics package, so it is necessary to load this before defining the function.

<< Graphics`

Here is the IntersectionPlot function and description.

This function is designed to allow the area common to two circles or ellipses to be shaded.  The form of the input is:

IntersectionPlot[{center1,radius1},{center2,rad2},plotrange1,plotrange2,opts]

where the radius can be given as a single number or an ordered pair of numbers.  In the second case the first number is the horizontal semi-axis and the second the vertical semi-axis.  The opts may be any standard options for Plot or InequalityPlot except DisplayFunction.

IntersectionPlot[{cent1_, rad1_}, {cent2_, rad2_}, plotrange1_, 
        plotrange2_, opts___?OptionQ] := 
      Module[{rng, func1, func2, plot1}, 
rng1  = 
          plotrange1 /. 
            plotrange1[[1]] ->  xx; 
rng2  = 
          plotrange2 /. 
            plotrange2[[1]] ->  yy; 
func1  = 
          If[NumberQ[
              rad1], (xx - cent1[[1]])^2  +  (yy - 
                      cent1[[2]])^2  == 
              rad1^2, ((xx - cent1[[1]])/
                      rad1[[1]])^2  +  ((yy - cent1[[2]])/
                      rad1[[2]])^2  == 
              1]; 
func2  = 
          If[NumberQ[
              rad2], (xx - cent2[[1]])^2  +  (yy - 
                      cent2[[2]])^2  == 
              rad2^2, ((xx - cent2[[1]])/
                      rad2[[1]])^2  +  ((yy - cent2[[2]])/
                      rad2[[2]])^2  == 
              1]; 
plot1  = 
          ImplicitPlot[{func1, func2}, rng1, 
            DisplayFunction  ->  Identity, 
            opts]; 
ineq1  = 
          If[NumberQ[
              rad1], (xx - cent1[[1]])^2  +  (yy - 
                      cent1[[2]])^2  <= 
              rad1^2, ((xx - cent1[[1]])/
                      rad1[[1]])^2  +  ((yy - cent1[[2]])/
                      rad1[[2]])^2  <= 
              1]; 
ineq2  = 
          If[NumberQ[
              rad2], (xx - cent2[[1]])^2  +  (yy - 
                      cent2[[2]])^2  <= 
              rad2^2, ((xx - cent2[[1]])/
                      rad2[[1]])^2  +  ((yy - cent2[[2]])/
                      rad2[[2]])^2  <= 
              1]; 
plot2  = 
          InequalityPlot[And[ineq1, ineq2], rng1, rng2, 
            DisplayFunction  ->  Identity, 
            opts]; 
plt  = 
          Show[{plot1, plot2}, 
            DisplayFunction  ->  $DisplayFunction]
]

Now let's use this function with our coordinates to get the overlapping segment.

\(overlap = 
      IntersectionPlot[{{0.425598, 0.603164}, 0.2}, {{0.565196, 0.698447}, 
          0.2}, 
{x, \(-2\), 5}, {yy, 0, 5}, 
        Fills\  -> RGBColor[1, 0, 1]]\)

This removes the lines so that all we are showing is the filled portion.

prims = First[overlap /. Line[_] -> {}];

Show[{Graphics[{RGBColor[1, 0, 0], Disk[{0.425598, 0.603164}, 0.2]}, 
          AspectRatio -> Automatic]}, {Graphics[{RGBColor[0, 0, 1], 
            Disk[{0.565196, 0.698447}, 0.2]}], Graphics[prims]}]

What is left to do:
You will need to create a function that will Map IntersectionPlot over your list of points.  You will also need to plot each red circle, and each blue circle.  Then you will have to show all three graphics together: the red circle, the blue circle, and the filled portion.  Please note that you can use CMYKColor instead of RGBColor, which may make it easier to specify which colors need to be shown.  I hope this gives you a helpful beginning to a very interesting problem.  I would be interested in seeing the final result if at all possible.  Also if you have any further questions, please let us know.