I have the following definitions:

```
a=Sqrt(2 + 2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)));
a3=Sqrt(5 + 4 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)));
beta = 2 a1^2 + 2 a2^2 + 2 a3^2 + 2 a1^2 a2^2 + 2 a1^2 a3^2 +
2 a2^2 a3^2 - a1^4 - a2^4 - a3^4 - Sqrt(((a + a + a3)^2 - 1)*((a - a + a3)^2 -
1)*((a + a - a3)^2 - 1)*((a - a - a3)^2 - 1)) - 1;
```

then, I define the plots `A`

and `B`

as follows:

```
A = Plot3D({1/2 Log((beta/(8 a^2)))}, {r, -1.0, 1.0}, {(Theta),
0.01 (Pi), 1.99 (Pi)},
RegionFunction ->
Function({r, (Theta)},
0 < Sqrt(
2 + 2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) -
1/Sqrt(2) ((Sqrt)(((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
2 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r))
Sqrt(Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) +
Abs(-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r))) (Sqrt)((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
8 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r))
Sqrt(Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r)))))/(7 +
6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))))),
PerformanceGoal -> "Quality", AxesLabel -> Automatic,
PlotRange -> All, PlotPoints -> 30, Mesh -> 5, MaxRecursion -> 7,
ColorFunction -> "TemperatureMap");
(*-------------------------------------------*)
B=Plot3D({1/2 Log(((a^2 - a3^2)/(a^2 - 1))^2)}, {r, -1.0,
1.0}, {(Theta), 0.01 (Pi), 1.99 (Pi)},
PerformanceGoal -> "Quality", AxesLabel -> Automatic,
RegionFunction ->
Function({r, (Theta)},
0 >= Sqrt(
2 + 2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) -
1/Sqrt(2) ((Sqrt)(((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
2 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) +
Abs(-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r))) (Sqrt)((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
8 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r))
Sqrt(Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r)))))/(7 +
6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))))),
PlotRange -> All, PlotPoints -> 30, Mesh -> 5, MaxRecursion -> 8,
ColorFunction -> "TemperatureMap");
```

then, by the command `Show()`

I join the two plots obtaining

Therefore, both 3Dplots **match**, as I expected; however, I have the question:

(1) **There is a way to show a uniform color distribution for both plots by using the comand** `Show()`

?

that is, each plot appears with its own color distribution when I display both with `Show()`

. This is logical since I define separately each function. On the other hand, it must be noted that the region function for plot `A`

is of the form: `RegionFunction -> Function({r, (Theta)}, 0 <f)`

and for `B`

is `RegionFunction -> Function({r, (Theta)}, 0 >=f)`

being `f`

the function of $r$ and $theta$ displayed in the code, which could help to define a conditional to display a single plot without the need to use `Show()`

.