# Need help with making an interactive 3D plot

My main problem is to generate a manipulatable 3D plot, combining three functions and two of those functions depend on another variable Theta (which shall be the manipulateable variable). To explain it in detail:

I have two functions, which depend on the variable Theta. The variable Theta shall later be the factor which can be manipulated. So at first I defined two functions (kbA and kbM), which depend on this variable:

No. 1)

CPA[ϕ_, α_, β_, δ_, x_, ρ_, θ_, kM_] :=
cA[t]*ϕ*(α kA^(α - 1) + β kM^(β - 1) - δ - ρ - x θ)/θ

kbA = kA /.  Flatten@Solve[
CPA[0.5, 1/3, 2/3, 0.05, 0.02, 0.01, θ, 500] == 0.0, kA]


No. 2)

CPM[ϕ_, α_, β_, δ_, x_, ρ_, θ_, kA_] :=
cM[t]*(1 - ϕ )*(α kA^(α - 1) + β kM^(β - 1) - δ - ρ - x θ)/θ

kbM = kM /. Flatten@Solve[CPM[0.5, 1/3, 2/3, 0.05, 0.02, 0.01, θ, 500] == 0.0, kM]


Those two functions (kbA and kbM) depend on Theta. (The results, kA* and kM*, of these functions theoretically define two coordinate values for a line plot I need later).

To generate the 3D plot I have a third function which looks like this:

KP[α_, β_, δ_, x_, n_] := -c[t] - (n + x + δ) ( kA + kM) + (kA^α +
kM^β)

cfun = c[t] /. Flatten @ Solve[KP[1/3, 2/3, 0.05, 0.02, 0.01] == 0.0, c[t]]

p3 = Plot3D[cfun, {kA, 0, 50}, {kM, 0, 1500}, AxesLabel -> {"Capital A \!$$\*SubscriptBox[OverscriptBox[\(k$$, $^\)], $$A$$]\)", "Capital M \!$$\*SubscriptBox[OverscriptBox[\(k$$, $$^$$], $$M$$]$", "Consumption  \!$$\*FormBox[OverscriptBox[\(c$$, $$^$$],TraditionalForm]\)"}]


Now I have my 3D plot p3 based on the function cfun. Into this 3D plot, I want to insert a lineplot, whose coordinates are the two results of KbA and kbM (and third coordinate is 1, because the line plot is parallel to the y-axis "consumption"). But kbA and kbM depend on Theta. And this variable Theta shall be manipulateable, in a graph.

But now I have the problem, to combine the results of kbA and kbM in a function or something else and to integrate it afterwards into the 3D plot! I don't know what I can do or should do, to solve my problem.. How can I say/define, that every value/result of kbA and KbM for the same value of Theta build the coordinates for the line plot i want to integrate in the 3D plot?

Can somebody help me? And even can tell me, how I can write the command/input for a mnipulation of Theta or rather of the final 3D plot?

So the main question is: How can I define/makte this line in the 3D plot manipulatable with respect to Theta? So that I change the value of Theta (i.e. with a control in the Manipulate expression) and the line moves inside the 3D plot?

I saw that the graphic wasn't shown. So here are the commands to solve it..

kbA3 = kbA3 /. Flatten@Solve[CPA[0.5, 1/3, 2/3, 0.05, 0.02, 0.01, 3.0, 500] == 0.0, kA]

kbM3 = kbM3 /. Flatten@Solve[CPM[0.5, 1/3, 2/3, 0.05, 0.02, 0.01, 3.0, 500] == 0.0, kM]

KP3[α_, β_, δ_, x_, n_, kA_, kM_] :=
-c[t] - (n + x + δ) ( kA + kM) + (kA^α + kM^β)

c3 = c[t] /. Flatten@NSolve[KP3[1/3, 2/3, 0.05, 0.02, 0.01, kbA3, kbM3] == 0.0, c[t]]

SS = Show[p3, Graphics3D @ { Line[{{kbA3, kbM3, 0}, {kbA3, kbM3, 24.0}}] } ]

• I want to help, but something is going wrong which others will not be able to fix for you. The way you have defined kbA3,kbM3 and KP3 in your question. just return those variables. I recommend you just give the exact form of these functions in your question, and leave out all the Solve bits. – Feyre Jun 11 '16 at 11:58

In order to perform the Manipulate that you are looking for we really only need three functions (see Feyre's comment).

My interpretation is that you could use:

kbA[θ_] := 6.80414/((θ/10 - 0.119974) Sqrt[-1.19974 + θ])
kbM[θ_] := 37037/(20.468 + 22.4478 θ + 8.2063 θ^2 + θ^3)
cfun[kA_, kM_] := kA^(1/3) + kM^(2/3) - 8 (kA + kM)/100


Now we can define your plot as:

p = Plot3D[cfun[kA, kM], {kA, 0, 50}, {kM, 0, 1500},
AxesLabel -> {"\!$$\*SubscriptBox[\(k$$, $$A$$]\)",
"\!$$\*SubscriptBox[\(k$$, $$m$$]\)",
"Consumption \!$$\*FormBox[OverscriptBox[\(c$$, $$^$$], I am not convinced that you have the right range for kA and kB but I will leave that to you.

Now let's look at a table of kbA and kbM.

Table[{θ, kbA[θ], kbM[θ]}, {θ,
Subdivide[0, 2 π, 10]}]

(* {{0, 0. + 51.7777 I, 1809.51}, {π/5, 0. + 157.521 I,
973.118}, {(2 π)/5, 5013.47, 582.16}, {(3 π)/5, 119.959,
375.492}, {(4 π)/5, 45.1972, 256.141}, {π, 25.1448,
182.458}, {(6 π)/5, 16.5132, 134.532}, {(7 π)/5, 11.8948,
102.023}, {(8 π)/5, 9.08904, 79.1987}, {(9 π)/5, 7.23574,
62.7052}, {2 π, 5.93658, 50.4912}} *)


Observe that low valued of θ result in an imaginary number.

I will plot the real component to take that into account but one could use the absolute value.

Here is the Manipulate

Manipulate[
Show[
p3,
Graphics3D[
{
Red,
Thick,
With[
{
kbAx = kbA[θ],
kbMx = kbM[θ]
},
Line[{{Re[kbAx], kbMx, 0}, {Re[kbAx], kbMx, 24.0}}]
]
}
]
],
{{θ, π}, 0, 2 π, Appearance -> "Open"}
] With the given plot ranges the red line will disappear from the plot for low and some intermediate values of θ.

Hope this gets you started. Good luck.