# Create a 2D region plot with 3 variables, 2 sharing the same axis

I have a region with 3 variables: x, s3 and s4. The value of s3 depends on x and the value of s4 depends on both s3 and x. The full region is as follows:

x > 45 && ((Inequality[0, Less, s3, LessEqual, (5/16)*(-55 + x) +
(5/16)*Sqrt[3025 + 330*x + x^2]] && 0 < s4 < (1/4)*(-275 - 4*s3 + 5*x) +
(1/4)*Sqrt[75625 + 2200*s3 + 16*s3^2 + 8250*x + 120*s3*x + 25*x^2]) ||
((5/16)*(-55 + x) + (5/16)*Sqrt[3025 + 330*x + x^2] < s3 < (5/24)*(-55 + 2*x) +
(5/24)*Sqrt[3025 + 440*x + 4*x^2] && (-1100*s3 - 32*s3^2 + 1375*x +
20*s3*x)/(4*s3 - 5*x) < s4 < (1/4)*(-275 - 4*s3 + 5*x) + (1/4)*Sqrt[75625 +
2200*s3 + 16*s3^2 + 8250*x + 120*s3*x + 25*x^2]))


I would like to plot this region in a single 2D graph, with the x-axis representing x and the y-axis representing both s3 and s4. I made a quick mockup of what the result should be like:

I have not been able to figure out how to do this. Does someone know the best way to approach this?

• Provide the region as copy/pastable Mathematica code, not as Latex. – MarcoB Jul 3 '18 at 18:52
• @MarcoB Sorry, changed it – meteoorkip Jul 3 '18 at 20:11
• I am afraid that I do not understand your question. What do you mean for an axis to represent both values? – MarcoB Jul 3 '18 at 21:38
• @MarcoB I want to plot the values of both s3 and s4 in a single graph. The issue is that the value of s4 is dependent on s3. I can plot it using a 3D region, but I would like to have a 2D plot showing both the value of s3 and s4. In other words, the y-axis should be used for both values – meteoorkip Jul 3 '18 at 22:18

As you can see from the RegionPlot3D plot, the $x-s3$ slices of the region depend on the value of s4 and the $x-s4$ slices of the region depend on the value of s3. So you can only get a 2D $x-s3$ region conditional on a value for s4 (similarly for $x-s4$ slices). This can be done along the following lines:

ClearAll[x, s3, s4, reg]
reg[x_, y_, z_] :=  x > 45 && ((0 < y <= 5/16 (-55 + x) + 5/16 Sqrt[3025 + 330 x + x^2] &&
0 < z < 1/4 (-275 - 4 y + 5 x) + 1/4 Sqrt[75625 + 2200 y + 16 y^2 +
8250 x + 120 y x + 25 x^2]) ||
(5/16 (-55 + x) + 5/16 Sqrt[3025 + 330 x + x^2] < y < 5/24 (-55 + 2 x) +
5/24 Sqrt[3025 + 440 x + 4 x^2] && (-1100 y - 32 y^2 +
1375 x + 20 y x)/(4 y - 5 x) < z < 1/4 (-275 - 4 y + 5 x) +
1/4 Sqrt[75625 + 2200 y + 16 y^2 + 8250 x + 120 y x + 25 x^2]));

Manipulate[Show[RegionPlot[reg[x, s3, a[[2]]], {x, 45, 500}, {s3, 0, 400},
FrameTicks -> {{All, All}, {Automatic, Automatic}},
FrameLabel -> {{Style[OverBar[S3], 16], Style[OverBar[S4], 16]}, {Style["x", 16], ""}},
BoundaryStyle -> Directive[Darker@Green, Thick],
PlotStyle -> Directive[Opacity[.5], Green],