computational geometry - Finding the perimeter, area and number of sides of a Voronoi cell

Does anyone have any suggestions how to determine the perimeter, area and number of sides of each Voronoi cell in Voronoi diagram?

Answer

Look at TileAreas in ComputationalGeometry:

Needs["ComputationalGeometry`"]
boundary = {{0, 0}, {10, 0}, {10, 10}, {0, 10}};
pts = RandomReal[{0, 10}, {100, 2}]
(Print[DiagramPlot[##]]; TileAreas[##]) & @@ 
 Prepend[BoundedDiagram[boundary, pts], pts]
(* {{0.261033,5.7592},{6.21362,4.0213},{4.44609,9.30305},
{7.10641,0.810209},{2.57901,9.65954},{2.34204,1.84401},

{7.76384,0.109391},{2.23168,6.84915},{5.59156,7.56046},{9.12543,8.8625}} *)

(* {6.3428,19.1094,4.97912,9.55327,5.70216,
17.6009,4.37766,10.6483,10.719,10.9674} *)

Simple Voronoi diagram

EDIT: Wait, you wanted perimeters too.

Function[{vert, adj},
  (Total[Norm /@ Subtract @@@ Partition[vert[[#[[2]]]], 2, 1, 1]]) & /@ 
   adj] @@ BoundedDiagram[boundary, pts]

(* {12.8885,17.3528,9.88682,14.4774,10.2263,16.1023,
      10.1097,13.5522,13.3678,14.3459} *)

SECOND EDIT: Number of sides

Length /@ BoundedDiagram[boundary, pts][[2, All, 2]]
(* {4, 6, 5, 5, 5, 6, 3, 6, 4, 5} *)

If you keep reusing the BoundedDiagram of many points, you should probably save it instead of recomputing each time like I'm doing.

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