MaplePrimes - answers and comments on Question, How to create Voronoi diagram with Maple?

Delaunay

acer — Fri, 10 Feb 2012 00:35:46 Z

If you had a graph for the Delaunay triangulation, would that help? (Ie, here.)

acer

Only in general position

Markiyan Hirnyk — Fri, 10 Feb 2012 00:58:24 Z

Up to Wiki http://en.wikipedia.org/wiki/Delaunay_triangulation#Relationship_with_the_Voronoi_diagram , the Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi tesselation for P. Special cases include the existence of three points on a line and four points on circle.

It is not so difficult

Markiyan Hirnyk — Sun, 12 Feb 2012 18:15:10 Z

Here is my code in the case n=3.
> restart; with(Statistics);
> with(plots);
> X := RandomVariable(Uniform(0, 10)); randomize();
> for j to 3 do A[j] := convert(Sample(X, 2), list) end do;
> cell[1] := proc (x, y) if is(sqrt((x-A[1][1])^2+(y-A[1][2])^2)-sqrt((x-A[2][1])^2+
(y-A[2][2])^2) <= 0) and is(sqrt((x-A[1][1])^2+(y-A[1][2])^2)-sqrt((x-A[3][1])^2+
(y-A[3][2])^2) <= 0) then 1 end if end proc;
> c1 := contourplot(cell[1], 0 .. 10, 0 .. 10, numpoints = 20000, contours = [1],
coloring = [blue, blue]);
> cell[2] := proc (x, y) if is(sqrt((x-A[2][1])^2+(y-A[2][2])^2)-sqrt((x-A[1][1])^2+
(y-A[1][2])^2) <= 0) and is(sqrt((x-A[2][1])^2+(y-A[2][2])^2)-sqrt((x-A[3][1])^2+
(y-A[3][2])^2) <= 0) then 1 end if end proc;
> c2 := contourplot(cell[2], 0 .. 10, 0 .. 10, numpoints = 20000, contours = [1],
coloring = [red, red]);
> c3 := pointplot([A[1], A[2], A[3]], color = black, symbol = solidcircle, symbolsize = 20);
> display({c1, c2, c3}, view = [0 .. 10, 0 .. 10]);

Frankly speaking, my picture leaves much to be desired so patronymic comments
and constructive suggestions are welcomed. There are a few Mathematica applications on this theme:
http://demonstrations.wolfram.com/VoronoiDiagrams/ ,
http://demonstrations.wolfram.com/LloydRelaxationOfVoronoiDiagrams/ ,
http://demonstrations.wolfram.com/VoronoiImage/ .
It should be noticed that my code is not a replica. I find it simpler to write this on my own.

Voronoi.mw

Another two versions

Markiyan Hirnyk — Wed, 15 Feb 2012 21:26:06 Z

restart; with(Statistics):
with(plots): N := 10;
setcolors(["Red", "LimeGreen", "Goldenrod", "Blue", "MediumOrchid", "DarkTurquoise",
"Brown", "Magenta", "Cyan", "Pink", "Gold"]);
C := setcolors();
X := RandomVariable(Uniform(0, 10)); randomize();
for j to N do A[j] := convert(Sample(X, 2), list) end do;
display(seq(plot3d(proc (x, y) if is(sqrt((x-A[k][1])^2+(y-A[k][2])^2)-
min(seq(sqrt((x-A[j][1])^2+(y-A[j][2])^2), `in`(j, `minus`({`$`(1 .. N)}, {k})))) <= 0)
then 1 end if end proc, 0 .. 10, 0 .. 10, style = surface, numpoints = 50000,
color = setcolors()[`mod`(k, nops(C))+1], orientation = [-90, 0]), k = 1 .. N),
pointplot3d([seq([op(A[j]), 1], j = 1 .. N)], symbol = solidsphere,
symbolsize = 10, color = black, orientation = [-90, 0]), scaling = constrained);

display(seq(implicitplot(sqrt((x-A[k][1])^2+(y-A[k][2])^2)-
min(seq(sqrt((x-A[j][1])^2+(y-A[j][2])^2), `in`(j, `minus`({`$`(1 .. N)}, {k})))) = 0,
x = 0 .. 10, y = 0 .. 10, thickness = 2, numpoints = 10000), k = 1 .. N),
pointplot([seq(A[j], j = 1 .. N)], symbol = solidcircle, symbolsize = 15),
scaling = constrained)

voronoi2.mw

Job well done

Markiyan Hirnyk — Tue, 22 Jan 2013 12:52:49 Z

I would like to circulate a great work by David Clark VoronoiDiagrams.mw which also is accessible through MapleCloud.

graphs, bisections, etc

acer — Fri, 16 Mar 2012 07:12:51 Z

Ok, I think I've got a better handle on understanding this now.

The Voronoi and Delaunay (incidence) graphs are dual, but that doesn't immediately give the geometric details of the Voronoi diagram (cells) from the details of the Delaunay triangulated mesh. But it turns out to not be so hard to compute.

I suppose that I am trying to say two main things:

1) One can either do the heavy lifting (distance calculations, like with Euclidean metric, involving circles) to compute the Delaunay triangulation, or the Voronoi diagram. But it shouldn't be necessary to make that effort for both. Now, I happen to have access to what seems to be a decent Delaunay triangulation mesh generator (producing details of the triangles and the incidence graph). I mostly only found schemes for computing the Voronoi diagram from scratch in various computational geometry texts. But a little observation reveals that the edges of the Voronoi cells may be computed directly from the intersections of the right bisectors of the triangles of the Delaunay triangulation.

2) We should be striving to compute the Voronoi cells as defined by straight line segments (well, according to choice of metric as Euclidean distance). So the program should be identifying those line segments. Therefore approaches (1, 2) which approximate those lines without final identification are not providing the ideal target.

I'm not saying that it's wrong to compute the Voronoi diagram the hard way. Whether to do "hard part" in computing the Voronoi or the Delaunay geometric details are a matter of choice. (And if you happen to have one then subsequently computing the other is much less hard.) But if one does elect to compute the Voronoi diagram as the hard part then the final result should be defining geometric details of the polytopes and not just a plot.

note: I will try to find time to produce a front-to-end computation: points -> Delaunay -> Voronoi cells, but I have only limited computer time this week.

acer