# Projection of a 4D region defined by function on 3D region

I am trying to project a 4D region defined by a function onto a 3D region so I could visualize it. I am trying to use Resolve to achieve this but it is not working. The code is shown below:

    ClearAll["Global*"]

regiondescriptor[x11_, x12_, x21_, x22_] :=
Tr[{{0.09, 0}, {0, 7}}.{{x11, x12}, {x21, x22}}] >= 1 &&
Tr[{{7, 0}, {0, 0.09}}.{{x11, x12}, {x21, x22}}] >= 1 &&
Tr[{{1.05, -0.95}, {-0.95, 1.05}}.{{x11, x12}, {x21, x22}}] >=1 &&
Tr[{{1.05, 0.95}, {0.95, 1.05}}.{{x11, x12}, {x21,x22}}] >= 1 &&
Min[Eigenvalues[{{x11, x12}, {x21, x22}}]] >= 0;

RegionPlot3D[Resolve[\!$$\*SubscriptBox[\(\[Exists]$$, $$x12$$]$${x11, x12, x21, x22} \[Element] regiondescriptor[x11, x12, x21, x22]$$\),Reals], {x11, -10, 10}, {x21, -10, 10}, {x22, -10, 10}]

• Somewhat related: (19575), (32844) – Michael E2 Jul 3 '16 at 15:06
• Do you expect all eigenvalues to be real? That is, if there are complex eigenvalues, should the point be considered outside the region? – Michael E2 Jul 3 '16 at 16:08

One way to try to "visualize" a 4D object is to take 3D hyperplane sections. A hyperplane is determined by a normal vector and a point. By varying each, one might construct a series that is informative. (One needs to have a good idea of what one is after, since there are many degrees of freedom and understanding the output is difficult.) I'll show how to do one or two sections.

The basic idea is to find a point preferably in the region so that the section will be nonempty in a neighborhood you can find. Pick a normal and complete it to an orthonormal basis we'll call coordsys. Basically, X == coordsys.Z + point, where X == {x11, x12, x21, x22} and Z = {z1, z2, z3, z4} are new coordinates. The hyperplane corresponds to the equation z1 == 0 or the parametrization coordsys.{0, z2, z3, z4} + point.

The OP's regiondescriptor emits error when the eigenvalues are complex. It's not clear whether such a point should be in the region or not. I rewrote the code so that they are excluded, but it can be adjusted to other criteria.

regiondescriptor[x11_, x12_, x21_, x22_] :=
Tr[{{0.09, 0}, {0, 7}}.{{x11, x12}, {x21, x22}}] >= 1 &&
Tr[{{7, 0}, {0, 0.09}}.{{x11, x12}, {x21, x22}}] >= 1 &&
Tr[{{1.05, -0.95}, {-0.95, 1.05}}.{{x11, x12}, {x21, x22}}] >= 1 &&
Tr[{{1.05, 0.95}, {0.95, 1.05}}.{{x11, x12}, {x21, x22}}] >= 1 &&
With[{eigs = Eigenvalues[{{x11, x12}, {x21, x22}}]},
Max@Im@eigs < 10^-12 && Min[Re@eigs] >= 0];


Random section:

SeedRandom;  (* for reproducibility *)
normal = Normalize@RandomReal[{-1, 1}, 4, WorkingPrecision -> 20];
point = RandomReal[{0, 10}, 4];
coordsys = Transpose@
N@DeleteCases[Orthogonalize[Join[{normal}, IdentityMatrix]], v_ /; v == {0, 0, 0, 0}];
normal = N@normal;


Check that the point is in the region:

regiondescriptor @@ point
(*  True  *)


Visualization of the section: The MeshFunctions correspond to the coordinates {x11, x12, x21, x22} to give some idea of the relationship of the section to the X coordinate system.

RegionPlot3D[regiondescriptor @@ (coordsys.{0, z2, z3, z4} + point),
{z2, -20, 80}, {z3, -40, 60}, {z4, -20, 80},
MeshFunctions ->
Function @@@ ({{z2, z3, z4}, #} & /@ (coordsys.{0, z2, z3, z4} + point)),
MeshStyle -> {Black, Cyan, Magenta, Yellow},
AxesLabel -> Automatic] Another random section through the same point (same RegionPlot code but with the domain {z2, -20, 80}, {z3, -60, 40}, {z4, -80, 20}):

normal = Normalize@RandomReal[{-1, 1}, 4, WorkingPrecision -> 20];
coordsys = Transpose@
N@DeleteCases[Orthogonalize[Join[{normal}, IdentityMatrix]], v_ /; v == {0, 0, 0, 0}];
normal = N@normal; A section orthogonal to the first coordinate axis (RotateLeft[] the matrix to get other axes), plotted over the domain {z2, -50, 50}, {z3, -60, 40}, {z4, -10, 90}.

coordsys = IdentityMatrix;
normal = coordsys[[All, 1]]; Update: Plotting the singular section

In response to a comment, here is an approach to plotting the points where the rank of the matrix {{x11, x12}, {x21, x22}} is less than 2. The idea is to use MeshFunctions to pick out the singular hypersurface Det[{{x11, x12}, {x21, x22}}] == 0. The problem in this case is that it appears to be a boundary of the 4D region. Since RegionPlot3D is discrete (very discrete), I had to fudge a little and pick out the subregion where -0.05 < Det[] < 0.05. (The determinant values are in the low thousands in most of the region, so this represents a relative error of 10^-4 or less.) The following is using the second example, "Another random section." (Of course to get reproducibility, you have to execute the first code again before executing the second.)

RegionPlot3D[regiondescriptor @@ (coordsys.{0, z2, z3, z4} + point),
{z2, -20, 80}, {z3, -60, 40}, {z4, -80, 20},
MeshFunctions -> {Function[{z2, z3, z4},
Det[{{x11, x12}, {x21, x22}}] /.
Thread[{x11, x12, x21, x22} -> (coordsys.{0, z2, z3, z4} + point)] // Evaluate]},
Mesh -> {{-0.05, 0.05}}, MeshShading -> {Magenta, Cyan},
PlotPoints -> 100, AxesLabel -> Automatic] I suspect that where the cyan is pinched (on the left, in three places) should show nodes (crossing lines). If so, that is an artifact of the fudging and the discretization.

• @ Michael E2 Thanks a lot for your thorough and detailed reply. I checked it and it is perfectly working. I just wonder if adding a Rank constraint to the regiondescriptor will work normally. I added MatrixRank[{{x11, x12}, {x21, x22}}] == 1 but not sure if what is being plotted, two intersecting lines, it correct or not. – Abdulrahman Kalbat Jul 4 '16 at 18:32
• @AbdulrahmanKalbat I would think it would work, since everything is being evaluated numerically, except for one thing. The == should drop the dimension of the region down by one, no? If so, the region would become a "hypersurface" (dim.-3 surface in R^4) and the hyperplane section will have dimension 2, i.e., an ordinary surface. I'm not sure how RegionPlot3D will react to that. (It might be easier to set Det[{{x11, x12}, {x21, x22}}] == 0, even though it captures rank 0, because it's continuous and plotting functions work better with continuous functions.) – Michael E2 Jul 4 '16 at 18:54
• @AbdulrahmanKalbat See my update for what I had in mind. – Michael E2 Jul 4 '16 at 19:27
• @ Michael E2 Thanks again for your detailed response. Was there any other part of the code changed. I can't reproduce this plot. Is cordsys is it random or the orthogonal one? – Abdulrahman Kalbat Jul 4 '16 at 20:34
• @AbdulrahmanKalbat It's the second example. I meant to include that in the post, but I guess I forgot. I added it to the answer. (To get the second one, you have to do SeedRandom and the normal and point for the first one before doing the second one.) Otherwise the only code I changed is the RegionPlot3D[] code. – Michael E2 Jul 4 '16 at 23:23

Since you can plot maximum in 3D, you can use the remaining variables as parameter. Here I use regiondescriptor as Opacity, so you can see spheres only in it returns true.

Manipulate[ Graphics3D[ Table[{Opacity[If[regiondescriptor[x11, x12, x21, x22],
1, 0 ] 0.5], Sphere[{x11, x12, x21}, .3]}, {x11, 0, 5}, {x12, 0, 5}, {x21, 0, 5}],
Axes -> True, AxesLabel -> {"x11", "x12", "x21"}], {x22, 1, 5, 1}] Or you can make a 2D plot using remaining two variables as parameter

Table[ListDensityPlot[Flatten[Table[{x11, x12,
If[regiondescriptor[x11, x12, x21, x22], 1, 0 ]}, {x11,5}, {x12, 5}], 1],
InterpolationOrder -> 0, PlotLabel -> {x21, x22}], {x21, 5}, {x22, 5}]// TableForm

• Thanks a lot for this nice solution. Is it possible to also consider negative values? Currently, it is only working for positive values. Also, is it possible to change the values of x_22 and plot all of the results on a single plot? That would be really great. – Abdulrahman Kalbat Jul 4 '16 at 18:49
• If regiondescriptor is returning negative values, then you can not use Opacity (unless you scale it within [0,1]), but you can use it as colour with proper scaling. You can plot multiple plots for different x2` and show them in a grid. – Sumit Jul 4 '16 at 19:36