The following code gets all vertices of all polygons (mesh cells) of VoronoiMesh[pts]:

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

This outputs:


My question

How can I get a list of vertices for each polygon and compute the area of each polygon using the Shoelace formula?

The output should be similar to:


So, by clicking on the polygon number, it should show its vertices and its size.

I found this tool-tip image in Finding the perimeter, area and number of sides of a Voronoi cell

  • 1
    $\begingroup$ Do you need to use the shoelace formula, or will the built in function Area suffice? $\endgroup$ – Chip Hurst Sep 8 '18 at 19:45
  • $\begingroup$ Yes. I need to use the shoelace formula, not built-in function. $\endgroup$ – Eman Sep 8 '18 at 20:54
  • $\begingroup$ See this for an implementation of the shoelace formula. $\endgroup$ – J. M.'s ennui Oct 12 '18 at 16:24
polygons = Join @@ MeshCells[mesh, 2, "Multicells" -> True][[All, 1]];
polygondata = With[{x = MeshCoordinates[mesh]}, Map[
    p \[Function] Partition[x[[p]], 2, 1, 1],
areas = 0.5 Total[Map[Det, polygondata, {2}], {2}];
circumferences = Total[Map[Norm, Differences /@ polygondata, {2}], {2}];

For the tooltipping, you can also use the option MeshCellLabel of MeshRegion, but that's are a bit unwieldy:

MeshRegion[mesh, MeshCellLabel -> Map[
   i \[Function] ({2, i} -> Tooltip[
         {"Vertices", polygons[[i]]},
         {"Vertex Coordinates", polygondata[[i, All, 1]]},
         {"Area", areas[[i]]},
         {"Perimeter", circumferences[[i]]}
        Alignment -> {Left, Top}
   Range[MeshCellCount[mesh, 2]]
  • $\begingroup$ Thanks so much for your help and your edit. That is helpful for getting the polygons' sizes of each polygon. If I want to show the vertices values of each polygon also. How can I do that?? Any suggestions?? $\endgroup$ – Eman Sep 8 '18 at 19:20
  • 1
    $\begingroup$ Have a look at the last edit. $\endgroup$ – Henrik Schumacher Sep 8 '18 at 19:28
  • $\begingroup$ Thanks so much for your help. I am really sorry for disturbance. But, I think the vertices in the code, gives the order of the vertices of each polygon, not the values of the vertices' points. $\endgroup$ – Eman Sep 8 '18 at 19:37
  • 1
    $\begingroup$ Is it better now? $\endgroup$ – Henrik Schumacher Sep 8 '18 at 19:42
  • 1
    $\begingroup$ You're welcome. $\endgroup$ – Henrik Schumacher Sep 8 '18 at 20:57

Use MeshPrimitives like this:

    0.666, 0.776, 0.952], 
     Grid@{{"Perimeter", Perimeter@p}, {"Area", Area@p}, {"Edges", 
        Length @@ p}}], {p, MeshPrimitives[mesh, 2]}]}], 
 Graphics[{Black, Point[pts], Red, Point[vertices]}]]

enter image description here

  • $\begingroup$ Thanks so much for your help. But, if I want the vertices of each polygon to be shown also with area,edges and Perimeter. How to do that?? $\endgroup$ – Eman Sep 8 '18 at 18:41
  • 1
    $\begingroup$ How are you ordering them? $\endgroup$ – M.R. Sep 8 '18 at 18:47
  • $\begingroup$ Thanks so much for your help and your reply. What did you mean by them ? Did you mean the vertices?? If you mean the vertices, I don't order them. the code get all vertices of all voronoi polygons. I want to get the vertices of each polygon, separately. So, by clicking on each polygon; I can get its vertices. $\endgroup$ – Eman Sep 8 '18 at 19:12
  • 2
    $\begingroup$ Note that PropertyValue[{mesh, 2}, MeshCellMeasure] is a faster way to get all of the areas. However I don't think the other properties can be computed in this way. $\endgroup$ – Chip Hurst Sep 8 '18 at 19:52

Here's an efficient way to implement the shoelace formula, assuming no self intersections:

ShoelaceArea[Polygon[pts_?MatrixQ]] := 
  0.5 * #1.(RotateLeft[#2] - RotateRight[#2])& @@ Transpose[pts]

A comparison:

shoeareas = ShoelaceArea /@ MeshPrimitives[mesh, 2]; // AbsoluteTiming
 {0.000233, Null}
areas = PropertyValue[{mesh, 2}, MeshCellMeasure]; // AbsoluteTiming
 {0.000013, Null}
Max[Abs[shoeareas - areas]]
  • $\begingroup$ Thanks so much for your help. $\endgroup$ – Eman Sep 8 '18 at 21:12

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