Best way to plot nearly identically overlapping data?

Question

I have two different models whose solutions in certain cases are virtually identical. For purposes of illustration, I'd like to plot both solutions in the same figure, but after playing around with Opactiy, Dashed, and Thick, the final result still appears simply as overlapping lines.

This post (Is it Possible to change dashes into circles with Plot command?) has interesting solutions with "dotting" instead of dashing, but these don't appear to show up in a plot legend.

Does Mathematica have a way of dealing with this, in a professional looking way?

As an example, take

F[x_] := 1/Sqrt[4 - x^2];
G[x_] := 1/Sqrt[3.99 - x^2];
Plot[{F[x], G[x]}, {x, 0, 2}, PlotLegends -> {"One", "Two"}]

Like the example, my curves are continuous, not discrete data points.

Answer 1

A rough example using plotGrid by Jens:

p1 = Plot[F[x], {x, 0, 1.9},
      PlotRange -> All, Epilog -> Inset["F(x)"]];

p2 = Plot[G[x] - F[x], {x, 0, 1.9},
      PlotRange -> All, Epilog -> Inset["G(x) - F(x)"]];

plotGrid[{{p1}, {p2}}, 300, 300]

Answer 2

Just another way on same plot...

Plot[{f[x], g[x], 100 (g[x] - f[x])}, {x, 0, 1.9}, Frame -> True, 
 FrameTicks -> {{Automatic, 
    Table[{j, j/100}, {j, 0, 2, 0.5}]}, {Automatic, None}}, 
 PlotStyle -> {Blue, Directive[Orange, Dashed], Red}, 
 FrameLabel -> {{Row[{Style["f(x)", Blue], " , ", 
      Style["g(x)", Orange]}], Style["g(x)-f(x)", Red]}, {"x", None}},
  PlotLegends -> {"f(x)", "g(x)", "g(x)-f(x)"}]

Answer 3

Be bold like Alexander the Great; cut the Gordian knot rather than untying it.

F[x_] := 1/Sqrt[4 - x^2]
Plot[F[x], {x, 0, 2},
  PlotLabel ->
    Column[
      {Style["F", Italic][x],
       Row[{G[x], " is nearly identical"}]},
      Alignment -> Left]]

For a more note-like appearance you can use Inset.

Plot[F[x], {x, 0, 2},
  PlotLabel -> Style["F", Italic][x],
  Epilog ->
    Inset[Row[{G[x], " is nearly identical"}], Scaled[{.8, .125}]]]

To further elucidate your work, you might add a difference plot to along with this one.

Answer 4

If you're plotting two very closely overlapping functions, it's usually acceptable to have them drawn over the top of each other. It could even be considered dishonest to manipulate the axes to exaggerate their differences.

To make a self-explaining plot, with two overlapping but clearly distinguishable lines, could you use PlotStyle, as shown here?: https://reference.wolfram.com/language/ref/PlotStyle.html

For example, you could make one line dashed and picked some appropriate colours.

Answer 5

Here's a neat way by slight modification of TwoAxisPlot.

TwoAxisPlot[{f_, g_}, {x_, x1_, x2_}] := 
 Module[{fgraph, ggraph, frange, grange, fticks, 
   gticks}, {fgraph, ggraph} = 
   MapIndexed[
    Plot[#, {x, x1, x2}, Axes -> True, 
      PlotStyle -> ColorData[1][#2[[1]]]] &, {f, g}]; {frange, 
    grange} = {{0.5, 1.5}, {0.52, 1.52}}; 
  fticks = N@FindDivisions[frange, 5];
  gticks = 
   Quiet@Transpose@{fticks, 
      ToString[NumberForm[#, 2], StandardForm] & /@ 
       Rescale[fticks, frange, grange]};
  Show[fgraph, 
   ggraph /. 
    Graphics[graph_, s___] :> 
     Graphics[
      GeometricTransformation[graph, 
       RescalingTransform[{{0, 1}, grange}, {{0, 1}, frange}]], s], 
   Axes -> False, Frame -> True, 
   FrameStyle -> {ColorData[1] /@ {1, 2}, {Automatic, Automatic}}, 
   FrameTicks -> {{fticks, gticks}, {Automatic, Automatic}}]]

Now our plot.

TwoAxisPlot[{F[x], G[x]}, {x, 0, 1.8}]