I am trying to create a geometric object represented by the intersection of a sphere and the space below a plane using ImplicitRegion. When I use the sphere alone, it works. When I use the plane alone, it also works... But when I use them both together, it does not work! I get the error message "DiscretizeRegion was unable to discretize the region ImplicitRegion".
(* A sphere by itself works *)reg1 = DiscretizeRegion[ ImplicitRegion[ x^2 + y^2 + z^2 == 1 && -40 <= x <= 40 && -40 <= y <= 40 && -20 <= z <= 40, {x, y, z}]];(* The region below a given plane alone works *)reg2 = DiscretizeRegion[ ImplicitRegion[-(64/(3 Sqrt[3])) + (16 x)/3 - (16 y)/3 + (16 z)/ 3 <= 0 && -40 <= x <= 40 && -40 <= y <= 40 && -20 <= z <= 40, {x, y, z}]];(* The sphere and region below the plane together does not work *)reg12 = DiscretizeRegion[ ImplicitRegion[ x^2 + y^2 + z^2 == 1 && -(64/(3 Sqrt[3])) + (16 x)/3 - (16 y)/3 + (16 z)/3 <= 0 && -40 <= x <= 40 && -40 <= y <= 40 && -20 <= z <= 40, {x, y, z}]];Show[reg1, reg2, reg12]Any idea how to make it work for the Implicit Region defined by both the sphere and the plane?