Four-way logarithmic plot

Question

Say I have a dataset named "d"

d = {{0.02`, 0.0012590355605140609`}, {0.07`, 
0.03190107785291465`}, {0.12000000000000001`, 
0.1341664962650996`}, {0.17`, 0.36075980746436004`}, {0.22`, 
0.357927357108827`}, {0.27`, 0.7918189679330538`}, {0.32000000000000006`, 
0.9908662217648183`}, {0.37000000000000005`, 
1.326284520883634`}, {0.42000000000000004`, 
1.6667182988224543`}, {0.47000000000000003`, 2.786860204849679`}, {0.52`, 
2.93933750379339`}, {0.5700000000000001`, 
3.9366944327584346`}, {0.6200000000000001`, 3.134060239061052`}, {0.67`, 
3.4525830444631516`}, {0.7200000000000001`, 3.146161177105212`}, {0.77`, 
3.020376751028269`}, {0.8200000000000001`, 
4.675746764366013`}, {0.8700000000000001`, 6.218831558416226`}, {0.92`, 
5.912788886349729`}, {0.9700000000000001`, 6.083549243252543`}, {1.02`, 
5.698685465342156`}, {1.07`, 5.335045431412702`}, {1.12`, 
4.588645123940877`}, {1.1700000000000002`,
  4.836149753666741`}, {1.2200000000000002`, 4.731911635811709`}, {1.27`, 
4.3204281272371095`}, {1.32`, 3.8191496326978123`}, {1.37`, 
4.425577764542655`}, {1.4200000000000002`, 
4.202433760690466`}, {1.4700000000000002`, 4.345090612510468`}, {1.52`, 
4.943409063693237`}, {1.57`, 5.5220625859488255`}, {1.62`, 
6.159173390833452`}, {1.6700000000000002`, 
6.429250552258077`}, {1.7200000000000002`, 6.3412110427206345`}, {1.77`, 
6.444125949011322`}, {1.82`, 6.941306477251247`}, {1.87`, 
7.269850600160908`}, {1.9200000000000002`, 
7.4459650057643145`}, {1.9700000000000002`, 7.381768092485144`}, {2.02`, 
7.638145357784952`}, {2.0700000000000003`, 8.126256325553179`}, {2.12`, 
8.057667374122504`}, {2.17`, 8.860957231660683`}, {2.22`, 
9.731314285903778`}, {2.27`, 10.521063318290198`}, {2.3200000000000003`, 
11.202904736346488`}, {2.37`, 11.746558133012728`}, {2.4200000000000004`, 
12.1887799898985`}, {2.47`, 12.524874206169914`}, {2.52`, 
12.947505240273822`}, {2.5700000000000003`, 13.340677305210736`}, {2.62`, 
13.590186938026871`}, {2.6700000000000004`, 14.14787161554347`}, {2.72`, 
15.664058697389326`}, {2.77`, 16.665468696630878`}, {2.8200000000000003`, 
17.000850998774546`}, {2.87`, 16.647131946787326`}, {2.9200000000000004`, 
16.3859238348108`}, {2.97`, 15.982010112784831`}};

and I generate two datasets from "d":

v = {#[[1]], #[[2]]*2*\[Pi]/#[[1]]} & /@ d;
a = {#[[1]], #[[2]]*(2*\[Pi]/#[[1]])^2} & /@ d;

I can plot "d", "v" and "a" individually as follows:

ListLinePlot[#] & /@ {d, v, a}

However I want to combine these three datasets into a 4-way logarithmic plot. The x-axis of the 4-way plot corresponds to the first column of the datasets i.e. period. The y-axis corresponds to the second column of the "v" dataset, the +45-degree axis corresponds to the second column of "d" dataset, and the +135-degree axis corresponds to the second column of the "a" dataset. This type of plot is used to present the response spectrum in earthquake engineering. Can anyone help with this type of plot? An example is shown.

Answer 1

Epilog and FullGraphics with RotationTransform do the main part of the visualization

shift = {1, -1} Log[2 π]/Sqrt[2];
scale = {1, 1}/Sqrt[2];
ao = {0.5, 50};
ListLogLogPlot[{0}, Axes -> False, 
 PlotRange -> {{0.01, 10}, {0.1, 100}}, Frame -> True, 
 GridLines -> Automatic, 
 FrameLabel -> {Style["x axis", 16], Style["v axis", 16]}, 
 AspectRatio -> Automatic, ImageSize -> 500,
 Epilog -> {
   GeometricTransformation[#, RotationTransform[π/4]] &@
         GeometricTransformation[#, TranslationTransform[shift]] &@
       GeometricTransformation[#, ScalingTransform[scale]] &@
     Join[
      List @@ FullGraphics@
        ListLogLogPlot[{0}, Axes -> True, AxesOrigin -> ao, 
         PlotRange -> {{0.0002, 200}, {0.1, 100000}}, 
         GridLines -> Automatic], {Text[Style["d axis", 16], 
        Log@ao + {0.7, -0.2} Log[10]], 
       Rotate[Text[Style["a axis", 16], 
         Log@ao + {-0.2, 0.7} Log[10]], -π/2]}] /. 
    Text[s__] :> Text[s, Background -> GrayLevel[1, 0.8]],
   AbsolutePointSize[6],
   ColorData[1][1], Point@Log@v
   }]

Some details:

• shift shifts the position of (1,1) point of the 45-degree plot.
• ao sets the axis origin of the 45-degree plot.
• Logs converts the logarithmic values to the linear Graphics coordinates.
• ColorData[1] sets colors to usual colors of Plot.
• /. Text[s__] :> Text[s, Background -> GrayLevel[1, 0.8]] set background of all text labels of the 45-degree plot to white with small transparency.
• Unfortunately it doesn't work in V10 due to bugs (in Linux at lest). I used V9 to plot it.