Discretize a Graphics3D object over a given 3D lattice

Question

Is there a way to discretize a Graphics3D object using a regular lattice with a given lattice spacing. For example:

Graphics3D[Sphere[3], PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}]

or in the case of multiple spheres:

ClearAll["Global`*"]
array1 = {1, 5, -2};
array2 = {2, 6, -1};
array3 = {3, 3, 2};
array4 = {1, 3, 2};
ONE = MapThread[
Ball[{#1, #2, #3}, #4] &, {array1, array2, array3, array4}];
Graphics3D[{Red, ONE}, ImageSize -> 400, ImageSize -> 600, 
PlotRange -> {{-10, 10}, {-10, 10}, {-10, 10}}, ImageSize -> 400, 
Axes -> True, 
LabelStyle -> {FontFamily -> "Arial", FontSize -> 20}];

I'd like 1s for lattice points lying in the sphere region and 0s for everything outside. For a lattice spacing of 1, I would expect 11*11*11 points.

Answer 1

I will answer your question using a more complex example based on two intersecting balls (solids -- spheres are surfaces).

Graphics3D[{Ball[{-2, 0, 0}, 3], Ball[{2, 0, 0}, 3]}, Boxed -> False]

3D region formed by the two balls.

 r = RegionUnion[{Ball[{-2, 0, 0}, 3], Ball[{2, 0, 0}, 3]}];

Regular lattice over the cube with corners at {{-5, -5, -5}, {5, 5, 5}} with lattice points separated by 0.25 in direction. (I want more points than a 11 x 11 x 11 lattice would give).

mesh = CoordinateBoundingBoxArray[{{-5, -5, -5}, {5, 5, 5}}, .25];
Dimensions @ mesh

{41, 41, 41, 3}

Selecting the points that lie within the region r.

pts = Select[Flatten[mesh, 2], RegionMember[r, #] &];
ListPointPlot3D[pts, 
  PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}},
  BoxRatios -> {1, 1, 1},
  ImageSize -> Medium]

Now let's build a boolean array with a 1 at each point of r that falls on the lattice.

sa = SparseArray @ Map[Boole[RegionMember[r, #]] &, mesh, {-2}]