Bounded polygons in Voronoi diagram and calculating the areas and the number of sides of each polygon

Question

I have some data as

data={{257.3`, 493.7`}, {43.666666666666664`,490.5`}, {111.91176470588235`,461.20588235294116`},{345.2142857142857`,460.5`}, {420.88461538461536`, 436.34615384615387`}, {318.1`,408.46`}, {277.`,400.7`}, {273.5`, 383.`}, {444.`,381.5`}, {208.28571428571428`,379.7857142857143`}, {510.9166666666667`,367.66666666666663`}, {584.7`, 366.`}, {301.125`, 355.5`}, {116.875`, 352.75`}, {423.14285714285717`,340.2142857142857`}, {360.2142857142857`,340.07142857142856`}, {234.7`, 318.1`}, {287.25`,303.25`}, {474.65`, 301.35`}, {110.71428571428571`, 299.5`}, {440.95714285714286`, 297.3`}, {536.4545454545455`,277.72727272727275`}, {321.`, 268.2142857142857`}, {439.3125`,249.875`}, {306.42857142857144`,242.78571428571428`}, {505.42857142857144`,242.10714285714286`}, {603.5370370370371`,217.3888888888889`}, {618.0909090909091`, 212.86363636363637`}, {248.5`, 212.875`}, {110.07894736842105`, 199.60526315789474`}, {384.7857142857143`, 188.`}, {572.8333333333334`,148.66666666666669`}, {33.`, 133.`}, {447.31481481481484`, 129.24074074074076`}, {206.33333333333334`, 116.41666666666669`}, {399.1764705882353`, 98.35294117647061`}, {33.3`,60.366666666666674`}, {216.875`, 44.`}, {328.77272727272725`, 40.31818181818181`}, {435.5`,38.5`}, {58.88461538461539`,37.11538461538464`}, {464.44117647058823`,23.323529411764696`}, {534.`, 1.375`}}

I would like to plot the Voronoi diagram of these data, so we have

Show[HighlightMesh[
VoronoiMesh[data], {Style[2, White], Style[1, Black]}], 
Graphics[{PointSize[Medium], Red, Point[data]}], Frame -> False, 
PlotRange -> Full]

which results in

I would like to calculate the number of sides of each polygon and also to calculate the area of them. But, before doing these stuff, how can I remove the unbounded polygons and only keep and work with the bounded ones?

Answer 1

vm = VoronoiMesh[data];

You can use the property "Interior" to get interior faces of vm:

interiorfaces = MeshPrimitives[vm, {2, "Interior"}];

Short[interiorfaces, 6]

interiorfaceindices = MeshCellIndex[vm, {2, "Interior"}]

HighlightMesh[vm, Style[interiorfaceindices, Red], 
 PlotTheme -> "Lines", BaseStyle -> Black]

Use Style[interiorfaceindices, Directive[EdgeForm[{Thick, Red}], FaceForm[]]] to get:

Construct a data set with edge counts, areas and region centroids of interiorfaces:

interiorfaceData = Map[Through@{Identity, Length@*First, Area, RegionCentroid}@# &]@
   interiorfaces;

Visualize interiorfaces using area for color-coding:

Graphics[{EdgeForm[LightGray],
   ColorData[{"Rainbow", MinMax @ interiorfaceData[[All, 3]]}] @ #3, #, 
   FontColor -> White, 
   Text[Style["Edge Count: "<>ToString @ #2, FontSize -> Scaled[.01]], #4, {0, -1}],
   Text[Style["Area: "<>ToString @ #3, FontSize -> Scaled[.01]], #4, {0, 1}]} & @@@
  interiorfaceData, ImageSize -> 800]

To get the edge counts and areas in a list:

edgecountsandareas = Map[Through@{Length@*First, Area} @ # &] @ interiorfaces

Answer 2

vm = VoronoiMesh[data];
poly = MeshPrimitives[vm, "Polygons"]

First evaluate the surrounding box

b = RegionBounds[vm]; 
rect =Line[{{b[[1, 1]], b[[2, 1]]}, {b[[1, 2]], b[[2, 1]]}, {b[[1, 2]],b[[2, 2]]}, {b[[1, 1]], b[[2, 2]]}, {b[[1, 1]], b[[2, 1]]}}];   

The intersection of rect and the polygons must be empty

rpoly=Select[poly, (RegionIntersection[#, rect] == EmptyRegion[2] &)]
Graphics[{rect,EdgeForm[Blue], FaceForm[White], rpoly}]

Number of sides of each polygon rpoly:

Map[Length@MeshPrimitives[# , "Line"] &, rpoly]  
(*{4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8}*)

Areas of each polygon rpoly:

Map[Area,rpoly]
(*{3200.36, 3344.58, 4752.63, 14360.1, 6804.33, 7685.5, 1993.66,  4454.98, 9498.88, 2834.43, 12996.1, 2909.38, 6107.07, 5160.02, 7882.94, 3504.96, 4228.52, 5680.42, 6216.63, 12875.9, 4052.56,  6972.53, 3428.06, 9326.52, 9019.31, 5562.72, 11252.}*)