mjhjmtu

The following code draws the polygons of VoronoiMesh[pts]:

SeedRandom[3]; 
pts = RandomReal[-1, 1, 25, 2];
mesh = VoronoiMesh[pts];
vertices = MeshCoordinates[mesh];
Show[mesh, Graphics[Black, Point[pts], Red, Point[vertices]]]

The output is:

The black points are pts, the red ones the vertices of the Voronoi mesh.

My questions:

• How can I draw circles around each point in pts with a given radius $r$, such as $0.18$?


• How can I compute <sum of circle areas>-<area of overlaps of the circles>, i.e.
  $$left(sum_cinmathrmCirclesmathrmArea(c)right)-left(sum_oinmathrmOverlapsmathrmArea(o)right)$$

I am not sure are you interested in the union of the disks or the union sans the intersections. The code below can be used for both cases.

Implicit regions

r = 0.18;
regs = ImplicitRegion[
 Sqrt[(x - #[[1]])^2 + (y - #[[2]])^2] <= r, x, y] & /@ pts;

Circles drawing

Show[mesh, 
 Graphics[Cyan, Circle[#, r] & /@ pts, Black, Point[pts], Red, 
 Point[vertices]], Axes -> True]

Union

c = Ceiling[Max[pts] + r, 0.5];

AbsoluteTiming[
 dregs = DiscretizeRegion[RegionUnion[regs], -2, 2, -2, 2, 
 ImageSize -> Medium]
 ]

RegionMeasure[dregs]

(* 1.99653 *)

Union Intersection

ires = DeleteCases[
 Flatten[Table[
 RegionIntersection[regs[[i]], regs[[j]]], i, 1, 
 Length[regs], j, i + 1, Length[regs]]], _EmptyRegion];

AbsoluteTiming[
 dires = DiscretizeRegion[RegionUnion[ires], -c, c, -c, c, 
 ImageSize -> Medium, Frame -> True, PlotRange -> -c, c, -c, c]
 ]

RegionMeasure[dregs] - RegionMeasure[dires]

(* 1.56288 *)

radius = .18;
disks = Disk[#, radius] & /@ pts;
25 Area[disks[[1]]]

2.54469

Area[RegionUnion[disks]]

2.03381

Show[mesh, Graphics[Black, Point[pts], Red, Point[vertices], 
 FaceForm[Opacity[.5,LightGreen]],EdgeForm[Thick,Darker@Green], disks]] 

I think it is much better to use RegionMeasure on the undiscretized regions. For instance:

disks = Disk[#, .18]& /@ pts;

ru = RegionMeasure @ RegionUnion[disks]
RegionMeasure @ DiscretizeRegion @ RegionUnion[disks]

2.03381

1.99723

The region measure of the discretized version is almost 2% off. Similarly:

ires = DeleteCases[
 Flatten @ Table[
 RegionIntersection[disks[[i]], disks[[j]]],
 i, 1, 25,
 j, i + 1, 25
 ],
 _EmptyRegion
];

int = RegionMeasure @ RegionUnion @ ires
RegionMeasure @ DiscretizeRegion @ RegionUnion @ ires

0.44757

0.439748

So a more accurate answer would be:

ru - int

1.58624

Finally, it is possible to get this result directly by using BooleanCountingFunction:

RegionMeasure @ BooleanRegion[
 BooleanCountingFunction[1, 25],
 disks
]

1.58624

although this version is slower than Antons.