RegionIntersection returns unevaluated for 3D regions

Question

I'm trying to find the volume an ellipsoid and a prism share. I tried using RegionIntersection, but it returns unevaluated. Input:

sample3D=Ellipsoid[{0, 0, 0}, {572, 96, 572}];
beam3D=Prism[{{2864.79, 0, -2810}, {0, 10.0001, -2810}, {-2864.79, 
   0, -2810}, {2864.79, 0, 2810}, {0, 10.0001, 2810}, {-2864.79, 0, 
   2810}}];
Volume@RegionIntersection[beam3D, sample3D]

Returns:

Volume[BooleanRegion[#1 && #2 &, {Prism[{{1145.93, 0, -2810}, {0, 
      10.0004, -2810}, {-1145.93, 0, -2810}, {1145.93, 0, 2810}, {0, 
      10.0004, 2810}, {-1145.93, 0, 2810}}], 
   Ellipsoid[{0, 0, 0}, {572, 96, 572}]}]]

Both the ellipsoid and prism are of the same dimension and are recognized by RegionQ as regions.

A two-dimensional example of a triangle and an ellipse works fine. Input:

sample2D=Disk[{0, 0}, {572, 10}];
beam2D=Triangle[{{2864.79, 0}, {0, 10.0001}, {-2864.79, 0}}];
Area@RegionIntersection[sample2D, beam2D]

Returns:

8927.06

I already read somewhere, that Mathematica versions below 11.3 had problems with Region functions, but I'm using version 12.0.

Thanks a bunch in advance!

Edit:

The original problem can be worked around by applying BoundaryDiscretizeRegion to both regions. Input:

sample3D$edit1 = 
  BoundaryDiscretizeRegion@Ellipsoid[{0, 0, 0}, {572, 96, 572}];
beam3D$edit1 = BoundaryDiscretizeRegion@
   Prism[{{2864.79, 0, -2810}, {0, 10.0001, -2810}, {-2864.79, 
      0, -2810}, {2864.79, 0, 2810}, {0, 10.0001, 2810}, {-2864.79, 0,
       2810}}];
Volume@RegionIntersection[beam3D$edit1, sample3D$edit1]

Returns:

9.37725*10^6

However, this only works for some points for Prism and not for others. Other points generate a bunch of error messages. Input:

sample3D$edit2 = 
  BoundaryDiscretizeRegion@
   Ellipsoid[{0, 0, 0}, {572, 96, 572}];
beam3D$edit2 = BoundaryDiscretizeRegion@
   Prism[{{21824., 11.9101, -2810}, {6823.95, 38.09, -2810}, {-15000, 
  0, -2810}, {21824., 11.9101, 2810}, {6823.95, 38.09, 2810}, {-15000,
   0, 2810}}];
Volume@RegionIntersection[beam3D$edit2, sample3D$edit2]

Returns error messages:

BoundaryDiscretizeRegion::regpnd: A non-degenerate region is expected at position 1 of BoundaryDiscretizeRegion[Prism[{{21824.,11.9101,-2810},{6823.95,38.09,-2810},{-15000,0,-2810},{21824.,11.9101,2810},{6823.95,38.09,2810},{-15000,0,2810}}]].
RegionIntersection::reg: BoundaryDiscretizeRegion[Prism[{{21824.,11.9101,-2810},{6823.95,38.09,-2810},{-15000,0,-2810},{21824.,11.9101,2810},{6823.95,38.09,2810},{-15000,0,2810}}]] is not a correctly specified region.
Volume::reg: RegionIntersection[BoundaryDiscretizeRegion[Prism[{{21824.,11.9101,-2810},{6823.95,38.09,-2810},{-15000,0,-2810},{21824.,11.9101,2810},{6823.95,38.09,2810},{-15000,0,2810}}]],] is not a correctly specified region.

Inspired by this answer I added Rationalize to the Prisms points and this works for most of my applications, albeit rather slowly. Input:

sample3D$edit3 := 
  BoundaryDiscretizeRegion@Ellipsoid[{0, 0, 0}, {572, 96, 572}];
beam3D$edit3 := 
  BoundaryDiscretizeRegion@
   Prism[Rationalize[{{150739.44950992631`, 
       23.691003428484592`, -2810}, {135739.44950992634`, 
       26.308997333058976`, -2810}, {-15000, 
       0, -2810}, {150739.44950992631`, 23.691003428484592`, 
       2810}, {135739.44950992634`, 26.308997333058976`, 
       2810}, {-15000, 0, 2810}}, 0]];
 Volume@RegionIntersection[beam3D$edit3, sample3D$edit3] // AbsoluteTiming

Returns:

{11.139, 486462.}

Eventhough this works, it is too slow for my ultimate goal and still returns error messages for some points:

BoundaryMeshRegion::bsuncl: The boundary surface is not closed because the edges Line[{{763,315},{316,763},{315,316}}] only come from a single face.

Answer 1

I managed to fix my problem with a workaround. It consists of two changes to my original problem:

First I split the Prism from my original problem into two prisms whose intersection formed the original prism. Second, I switched from BoundaryDiscretizeRegion to BoundaryDiscretizeGraphics, which works a lot faster. I also added MaxCellMeasure->0.05 to my Ellipsoid. Both higher and lower values for MaxCellMeasure give the error message:

BoundaryMeshRegion::bsuncl :  The boundary surface is not closed because the edges Line[...] only come from a single face.

But 0.05 seems to be a "sweetspot" and all calculations for my application work with this value. The now working piece looks like this:

incPrism := 
  BoundaryDiscretizeGraphics[
   Prism[Rationalize[Join[incPrismPtsNegative, incPrismPtsPositive], 
     0]]];
outPrism := 
  BoundaryDiscretizeGraphics[
   Prism[Rationalize[Join[outPrismPtsNegative, outPrismPtsPositive], 
     0]]];
sample3D := 
 BoundaryDiscretizeGraphics[
  Ellipsoid[{0, 0, 0}, {SampleWidth/2, SampleHeight, SampleWidth/2}], 
  MaxCellMeasure -> 0.05]
ParallelTable[
 Volume@RegionIntersection[outPrism, sample3D], {alpha, 
  0.01 \[Degree], 0.2 \[Degree], 0.01 \[Degree]}]
ParallelTable[
 Volume@RegionIntersection[incPrism, sample3D], {alpha, 
  0.01 \[Degree], 0.2 \[Degree], 0.01 \[Degree]}]

With SampleWidth and SampleHeight being non-negative numbers for the ellipsoids radii, incPrismPtsNegative, incPrismPtsPositive, outPrismPtsNegative and outPrismPtsPositive being the corresponding points for incPrism and outPrism, which are dependant on the angle alpha, which is used in ParallelTable. Returns:

{2.63661*10^6, 5.26535*10^6, 7.88319*10^6, 1.04872*10^7, 1.30742*10^7,
  1.56414*10^7, 1.81854*10^7, 2.07027*10^7, 2.31899*10^7, 
 2.56421*10^7, 2.80515*10^7, 3.04142*10^7, 3.27272*10^7, 3.49878*10^7,
  3.71931*10^7, 3.93402*10^7, 4.14256*10^7, 4.34455*10^7, 
 4.53931*10^7, 4.55803*10^7}
{4.55883*10^7, 4.55883*10^7, 4.55882*10^7, 4.5588*10^7, 4.55878*10^7, 
 4.55876*10^7, 4.55873*10^7, 4.5587*10^7, 4.55867*10^7, 4.55863*10^7, 
 4.55859*10^7, 4.5585*10^7, 4.55849*10^7, 4.55844*10^7, 4.55838*10^7, 
 4.55832*10^7, 4.55825*10^7, 4.55818*10^7, 4.55811*10^7, 4.55803*10^7}

Thanks a lot for your comments everyone!

Edit

With further tinkering I found that MaxCellMeasure->0.05 is not the be-all and end-all to my problem. However, playing around with the value for MaxCellMeasure gives satisfactory results most of the time. For my application I still count this as a success.