RegionUnion works but RegionIntersection does not

Question

I am trying to create an Alberti's window figure as described in this question, in which a (three-dimensional) polyhedron (e.g., dodecahedron) is projected onto a (two-dimensional) plane. I realized that the previous approach and solution of projecting points is not quite what I need. Instead, I'd like to project two-dimensional faces (in this case pentagons). Everything works except the final stage. Here's how I proceed:

1) Here are the vertices of the dodecahedron:

myVertices = N@PolyhedronData["Dodecahedron", "Vertices"];

2) Here are the grouped indices of each pentagon face:

myFaces = PolyhedronData["Dodecahedron", "Faces"];

3) Here are the selected faces that are visible from the center of projection:

mySelectedFaces = myFaces[[#]] & /@ {2, 3, 5, 6, 7};

4) Here is the center of projection:

cop = {10, 0, 0};

5) Here are the three-dimensional cones defined by a pentagon face on the dodecahedron and the center of projection:

myCones = Join[#, {cop}] & /@ (myVertices[[#]] & /@ mySelectedFaces);

6) Here's the region mesh of just the first such cone:

myConeMesh = ConvexHullMesh[myCones[[1]]]

7) I'm projecting onto a plane defined by:

poly = Polygon[{{6, -2, -2}, {6, -2, 2}, {6, 2, 2}, {6, 2, -2}}];

8) If I create the union of the cone and the projection plane, I get just what I expect:

RegionUnion[myConeMesh, poly]

9) But of course I want instead the intersection of the cone and the projection plane. That should give me a pentagon floating in the plane of the projection plane. (I could then color it, render it however I wish, and so forth.) However, when I implement what I think should be the obvious function, I do not get the desired intersection:

RegionIntersection[myConeMesh, poly]

I have tried all manner of putting the elements in braces, creating Mesh or ConvexHullMesh, etc., without success. I thought the problem might stem from the embedding dimension, but both component regions are in three dimensions.

How can I compute the pentagon intersection region within the projection plane?

Answer 1

Here's one workaround.

Since your clip-plane is axes aligned, we can use the 2 argument form of DiscretizeRegion to clip the solid in one direction, then manually pick the faces on the plane.

(If your clip-plane is not axes aligned, you could rotate your whole scene, clip, then rotate back.)

x = 6.;
clip = BoundaryDiscretizeRegion[myConeMesh, {{-1, x}, {-1, 1}, {-1, 2}}];

int = MeshRegion[
 MeshCoordinates[clip], 
 Pick[MeshCells[clip, 2], PropertyValue[{clip, 2}, MeshCellCentroid][[All, 1]], x]
];

Show[
 MeshRegion[int, BaseStyle -> ColorData[112, 1], PlotTheme -> "Minimal"],
 BoundaryMeshRegion[myConeMesh, BaseStyle -> Opacity[.3], PlotTheme -> "Minimal"]
]