# ListDensityPlot3D showing empty plot [closed]

Consider the following crudely implemented function

f[\[Phi]1_, \[Phi]2_, \[Phi]3_] :=
Module[{r, a1, a2, a3, a4, w1, w2, w3, w4, basis4D},
r = {Cos[\[Phi]1], Sin[\[Phi]1] Cos[\[Phi]2],
Sin[\[Phi]1] Sin[\[Phi]2] Cos[\[Phi]3],
Sin[\[Phi]1] Sin[\[Phi]2] Sin[\[Phi]3]};
r = r/Plus @@ r // N;

a1 = {1, 0, 0, 0};
a2 = {0, 1, 0, 0};
a3 = {0, 0, 1, 0};
a4 = {0, 0, 0, 1};

w1 = Normalize[a1 - a2];
w2 = Normalize[a1 - a3];
w2 = Normalize[w2 - (w1 . w2) w1];
w3 = Normalize[a1 - a4];
w3 = Normalize[w3 - (w1 . w3) w1 - (w2 . w3) w2];
w4 = Table[
Sum[
LeviCivitaTensor[4][[i, j, k, l]] w1[[j]] w2[[k]] w3[[l]], {j,
1, 4}, {k, 1, 4}, {l, 1, 4}], {i, 1, 4}] // Normalize;
basis4D = {w1, w2, w3, w4};
(r . # & /@ basis4D)[[1 ;; 3]]
]


Maps points $$(\phi_1,\phi_2,\phi_3)$$, $$\phi_i\in[0,\frac{\pi}2]$$, on a region of a 3-sphere (embedded in 4-dimensions) to points $$(x,y,z)$$ on a tetrahedron in 3D. And similarly

g[X_] := Module[{\[Phi]1, \[Phi]2, \[Phi]3, val, x, y, z},
{\[Phi]1, \[Phi]2, \[Phi]3, val} = X;
{x, y, z} = f[\[Phi]1, \[Phi]2, \[Phi]3];
{x, y, z, val}
]


mapping $$(\phi_1,\phi_2,\phi_3,val)$$ to $$(x,y,z,val)$$. To see how points are mapped, for example see

f[#[[1]], #[[2]], #[[3]]] & /@
N@Tuples[Range[0, \[Pi]/2, (\[Pi]/2)/8], 3] // ListPointPlot3D


Now, if I use this with ListDensityPlot3D it returns an empty plot. Say

data = Flatten[#, 2] &@
Table[{x, y, z, x y z // N}, {z, 0, \[Pi]/2, \[Pi]/2/10}, {y,
0, \[Pi]/2, \[Pi]/2/10}, {x, 0, \[Pi]/2, \[Pi]/2/10}];
ListDensityPlot3D[data] # Works fine

dataTransformed = ParallelMap[g, data // N];
ListDensityPlot3D[dataTransformed] # Empty plot


Why is that the case? Is there a way to fix that? If I do the same in one lower dimensions, namely project some data on one quadrant of a 2-sphere to a 2-simplex (triangle) and use ListDensityPlot, then everything works fine.

• Observation: data vs dataTransformed looks like this.
– Syed
May 21, 2022 at 3:36
• I’m voting to close this question because the OP has not provided any feedback for an extended period of time.
– Syed
Oct 12, 2023 at 10:04