Parametric Plotting 6D Dimensions to 3D for Visualizing a Solution

Question

I am trying to find out when: $$f(x,y,z,α,β,γ):= z\cos(β)\cos(γ)+y(\cos(γ)\sin(α)\sin(β)+\cos(α)\sin(γ))+x(−\cos(α)\cos(γ)\sin(β)+\sin(α)\sin(γ)),$$ where $α,β, \text{and } γ$ can be any value in $[-2\pi,2\pi]$ is positive.

But solving this is very difficult, and I don't know if it can be solved exactly in Mathematica without having more insights. (It comes from rotating the point $\{x,y,z\}$ using the RollPitchYawMatrix in Mathematica, and then taking the last coordinate.)

So, I thought about parametric plotting: $$\{x(−\cos(α)\cos(γ)\sin(β)+\sin(α)\sin(γ)), y(\cos(γ)\sin(α)\sin(β)+\cos(α)\sin(γ)), z\cos(β)\cos(γ)\},$$ while making the angles $α,β, \text{and } γ$ as RGBColors with $α$ being how red, $γ$ being how green, and $β$ being how blue with the opacity being 1 when $f()$ is positive, and 0 if it is non-positive. For points that can be represented by more than one way, the color would be the average of the colors that give that point. (For example gray is the average of black and white.)

The problem I have is that ParatmetricPlot3D in Mathematica is of the form:

ParametricPlot3D[{f[u, v], g[u, v], h[u, v]}, {u, u_min, u_max}, {v, v_min, v_max}]

But I ideally want something like:

ParametricPlot3D[{x*f[t, u, v], y*g[t,u, v], z*h[t,u, v]}, {x,x_min, x_max}, {y,y_min, y_max},{z,z_min, z_max}, {t, t_min, t_max}, {u, u_min, u_max}, {v, v_min, v_max}].

But that is not allowed.

I also started trying something like this:

ParametricPlot3D[
 {(-Cos[α] Cos[γ] Sin[β] + Sin[α] Sin[γ]),
  (Cos[γ] Sin[α] Sin[β] + Cos[α] Sin[γ]),
  Cos[β] Cos[γ]},
 {{α, -2 π, 2 π},
  {β, -2 π, 2 π},
  {γ, -2 π, 2 π}},
 ColorFunction -> Function[{x, y, z, α, β, γ},
   RGBColor[(α + 2 π)/(4 π),(γ + 2 π)/(4 π),(β + 2 π)/(4 π)]
  ],
 ColorFunctionScaling -> False
],

but I could not figure out how to actually fix it.

My thought is that for a fixed domain, the solution should appear to look like a 3D ribbon with thickness curved in a quasi-periodic way, with a periodic color scheme. But there is a good chance that I am wrong.

Answer 1

Nothing fancy it is just a plane for particular values of $α, β, γ$. On one side of the plane the function is negative on the other positive and right at the plane it is zero.

Manipulate[
 ContourPlot3D[
  z Cos[β] Cos[γ] + 
    y (Cos[γ] Sin[α] Sin[β] + 
       Cos[α] Sin[γ]) + 
    x (-Cos[α] Cos[γ] Sin[β] + 
       Sin[α] Sin[γ]) == 0, {x, -2, 2}, {y, -2, 
   2}, {z, -2, 2}], {α, 0, 2 π}, {β, 0, 
  2 π}, {γ, 0, 2 π}]

Or you can depict it the other way around. Manipulate x,y,z and plot in in some region defined by α, β, γ.

Manipulate[
 ContourPlot3D[
  z Cos[β] Cos[γ] + 
    y (Cos[γ] Sin[α] Sin[β] + 
       Cos[α] Sin[γ]) + 
    x (-Cos[α] Cos[γ] Sin[β] + 
       Sin[α] Sin[γ]) == 0, {α, 0, 
   2 π}, {β, 0, 2 π}, {γ, 0, 2 π}], {x, -2, 
  2}, {y, -2, 2}, {z, -2, 2}]

Or if you want to see precisely where it is negative and where positive you could use RegionPlot3D instead.

Manipulate[
 RegionPlot3D[{z Cos[β] Cos[γ] + 
    y (Cos[γ] Sin[α] Sin[β] + 
       Cos[α] Sin[γ]) + 
    x (-Cos[α] Cos[γ] Sin[β] + 
       Sin[α] Sin[γ]) < 0, 
   z Cos[β] Cos[γ] + 
    y (Cos[γ] Sin[α] Sin[β] + 
       Cos[α] Sin[γ]) + 
    x (-Cos[α] Cos[γ] Sin[β] + 
       Sin[α] Sin[γ]) > 0}, {x, -2, 2}, {y, -2, 
   2}, {z, -2, 2}, PlotStyle -> {Automatic, Opacity[0.5]}], {α,
   0, 2 π}, {β, 0, 2 π}, {γ, 0, 2 π}]

Manipulate[
 RegionPlot3D[{z Cos[β] Cos[γ] + 
    y (Cos[γ] Sin[α] Sin[β] + 
       Cos[α] Sin[γ]) + 
    x (-Cos[α] Cos[γ] Sin[β] + 
       Sin[α] Sin[γ]) < 0, 
   z Cos[β] Cos[γ] + 
    y (Cos[γ] Sin[α] Sin[β] + 
       Cos[α] Sin[γ]) + 
    x (-Cos[α] Cos[γ] Sin[β] + 
       Sin[α] Sin[γ]) > 0}, {α, 0, 
   2 π}, {β, 0, 2 π}, {γ, 0, 2 π}, 
  PlotStyle -> {Automatic, Opacity[0.5]}], {x, -2, 2}, {y, -2, 
  2}, {z, -2, 2}]