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    $\begingroup$ Indeed, your result gives the floating pentagon, as needed (thanks), but this is such a kludgy workaround. I will be performing lots of these kinds of projections—onto planes of arbitrary orientation—that rotating, clipping, and so forth is simply unwieldy. Do you have any insight as to why the straightforward RegionIntersection doesn't work? $\endgroup$ Commented Aug 19, 2019 at 18:58
  • $\begingroup$ Indeed a messy workaround. I don't really know why RegionIntersection doesn't work, but note that RegionUnion simply places both region into the same scene and doesn't resolve their intersections. You can see this with RegionUnion[myConeMesh, poly] // ConnectedMeshComponents. RegionIntersection on the other hand must find the intersections. $\endgroup$ Commented Aug 19, 2019 at 19:02
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    $\begingroup$ I figured out a better workaround: Define poly = Cuboid[{5.9,-2,-2},{6,2,2}]. This three-dimensional region then works. The earlier problem must be due to something about intrinsic dimension. $\endgroup$ Commented Aug 19, 2019 at 19:09
  • $\begingroup$ Technically that's not a plane, but you could then manually pick out the faces like in my answer. Also BoundaryDiscretizeRegion[myConeMesh, {{x - .001, x}, {-1, 1}, {-1, 2}}] works like this too. $\endgroup$ Commented Aug 19, 2019 at 19:12
  • $\begingroup$ Anyway.... thanks for your help (+1)... but not a full accept, since I have a sense someone will see through the fundamental problem and come to a true planar solution. $\endgroup$ Commented Aug 19, 2019 at 19:17