Viewing the inside of cuboid intersections

Question

   
I have a number of cuboids intersecting as illustrated. I would like to see their intersection, almost the view from inside. Setting the ViewPoint to {0,0,0} does not reveal much of the structure. Is there some way to visualize this with clever Graphics[] options, or must I compute the convex polyhedron that is the intersection of the cuboids (which is really what I want to see)?

Solved by Jens:

Answer 1

It can in principle be done using RegionPlot3D. But to get sharp edges, one needs to crank up the number of PlotPoints. Here is an example with four cubes that doesn't take too long to plot:

n = 4;

insideCube[pt_, l_] := And @@ Thread[Abs[pt] < l]

rotations = 
 MapThread[
  RotationMatrix[{{0, 0, 1}, {Cos[#1] Sin[#2], Sin[#1] Sin[#2], 
      Cos[#2]}}] &, RandomReal[{0, Pi}, {2, 4}]]

(*
==> {{{0.701529, 0.148251, -0.69705}, {0.148251, 0.926364, 
   0.346226}, {0.69705, -0.346226, 0.627892}}, {{0.989833, 
   0.065586, -0.12621}, {0.065586, 0.576913, 
   0.814168}, {0.12621, -0.814168, 0.566746}}, {{0.999495, -0.0316019,
    0.00319107}, {-0.0316019, -0.979314, 
   0.199866}, {-0.00319107, -0.199866, -0.979818}}, {{0.18502, 
   0.225404, -0.956536}, {0.225404, 0.937659, 
   0.264555}, {0.956536, -0.264555, 0.122679}}}
*)

rotatedCubes = 
 Or @@ MapThread[
   insideCube[#1.{x, y, z}, #2] &, {rotations, 
    RandomReal[{1, 1.1}, n]}]

(*
==> (Abs[0.701529 x + 0.148251 y - 0.69705 z] < 1.02814 && 
   Abs[0.148251 x + 0.926364 y + 0.346226 z] < 1.02814 && 
   Abs[0.69705 x - 0.346226 y + 0.627892 z] < 
    1.02814) || (Abs[0.989833 x + 0.065586 y - 0.12621 z] < 1.04828 &&
    Abs[0.065586 x + 0.576913 y + 0.814168 z] < 1.04828 && 
   Abs[0.12621 x - 0.814168 y + 0.566746 z] < 
    1.04828) || (Abs[0.999495 x - 0.0316019 y + 0.00319107 z] < 
    1.0917 && Abs[-0.0316019 x - 0.979314 y + 0.199866 z] < 1.0917 && 
   Abs[-0.00319107 x - 0.199866 y - 0.979818 z] < 
    1.0917) || (Abs[0.18502 x + 0.225404 y - 0.956536 z] < 1.02296 && 
   Abs[0.225404 x + 0.937659 y + 0.264555 z] < 1.02296 && 
   Abs[0.956536 x - 0.264555 y + 0.122679 z] < 1.02296)
*)

RegionPlot3D[rotatedCubes, {x, -2, 2}, {y, -2, 2}, {z, -2, 
  2}, PlotPoints -> 120, Mesh -> False]

To cut through the shape and see the inside, you'd have to add another condition to RegionPlot3D or you can restrict the PlotRange as in this example:

Show[%, PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}, {-1.5, 0}}]