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Consider the expression

RandomPoint@
 BooleanRegion[#1 && ! #2 &, {Sphere[{0, 0, 0}], ConicHullRegion[{{0, 0, 0}}, {{1, 1, 1}}]}]

This does not evaluate to a random point in the region (in 12.3), and instead stays unevalauted. However, the argument is both RegionQ and ConstantRegionQ, as required by the documentation:

RandomPoint can generate random points for any RegionQ region that is also ConstantRegionQ.

At first I thought it might be due to the degeneracy of the example, but using RandomPoint @ BooleanRegion[#1 && ! #2 &, {Sphere[{0, 0, 0}], ConicHullRegion[{{0, 0, 0}}, {{1, 1, 1}, {0, 1, 1}, {1, 0, 1}}]}] fails as well. (Yet this BooleanRegion is bona fide: try putting Region in front.)

It doesn't seem to be a problem with BooleanRegion, as e.g. RandomPoint @ BooleanRegion[#1 && ! #2 &, {Sphere[], Ball[{1,1,1}]}] works.

What's happening here, and what's a suitable workaround?

EDIT: Looks like applying DiscretizeRegion before RandomPoint works! But why doesn't RandomPoint evaluate as expected?

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  • $\begingroup$ In short it doesn’t know how to find them for this region so it comes back unevaluated. Discretizing gives a different region (but close approximation) and RandomPoint knows how to work with meshes. $\endgroup$
    – Greg Hurst
    Aug 1, 2021 at 1:30

1 Answer 1

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Update

I found an example to show the limitation of RegionIntersection.

reg = RegionIntersection[Circle[{0, 0}, {5, 4}], 
   Circle[{0, 0}, {2, 8}]];
(* EmptyRegion[2] *)
RandomPoint[reg]
dreg = RegionIntersection[DiscretizeRegion@Circle[{0, 0}, {5, 4}], 
   DiscretizeRegion@Circle[{0, 0}, {2, 8}]];
RandomPoint[dreg]

RandomPoint[EmptyRegion[2]]

{-1.76789, 3.73945}

Original

Maybe ConicHullRegion is numeric and Sphere is symbolic, so they need to coordinate. The last result seems depend on the structure of discretize region.

SeedRandom[1];
BooleanRegion[#1 && ! #2 &, {Sphere[{0, 0, 0}], 
   Line[{{0, 0, 0}, {1, 1, 1}}]}] // RandomPoint
RegionDifference[Sphere[], 
  Triangle[{{0, 0, 0}, {1, 1, 1}, {2, 1, 1}}]] // RandomPoint
RegionDifference[Sphere[], 
   ConicHullRegion[{{0, 0, 0}}, {{1, 1, 1}, {2, 1, 1}}]] // 
  DiscretizeRegion // RandomPoint

{0.245529, -0.968234, 0.0473094}

{-0.528872, 0.54312, -0.652162}

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