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(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     26539,        959]*)
(*NotebookOutlinePosition[     27430,        989]*)
(*  CellTagsIndexPosition[     27386,        985]*)
(*WindowFrame->Normal*)



Notebook[{

Cell[CellGroupData[{
Cell["All k-th order Voronoi diagrams, simultanuously.", "Subtitle",
  PageWidth->PaperWidth],

Cell[CellGroupData[{

Cell["Code", "Section"],

Cell["\<\
<<DiscreteMath`Combinatorica`
Off[General::spell1];
<<Graphics`Colors`
<<DiscreteMath`ComputationalGeometry`
<<VertexEnum.m
<<FaceLattice.m

ca[cl_] := Module[
\t\t(* Determinants for circle center *)
        {m,i},
        m = Table[Append[cl[[i]],1],{i,1,3}];
        Det[m]
]
cd[cl_] := Module[
        {m,i},
         m = 
      Table[{Plus @@{cl[[i]][[1]]^2, cl[[i]][[2]]^2}, cl[[i]][[2]], 1},
                {i, 1, 3}];
        -Det[m]
]
ce[cl_] := Module[
        {m,i},
         m = 
      Table[{Plus @@{cl[[i]][[1]]^2, cl[[i]][[2]]^2}, cl[[i]][[1]], 1},
                {i, 1, 3}];
        Det[m]
]
cf[cl_] := Module[
        {m,i},
         m = Table[{Plus @@{cl[[i]][[1]]^2, cl[[i]][[2]]^2}, cl[[i]][[1]],
                 cl[[i]][[2]]}, {i, 1, 3}];
        -Det[m]
]


circlecenter[xyz_List] := Module[
\t\t\t(* computes the centre of the circle, defined by the points <xyz> *)
\t\t{a,d,e},
\t\ta = ca[xyz];
\t\tIf[ a==0,
\t\t\tPrint[\"Dividing by zero\"];
\t\t];
\t\td = cd[xyz];
\t\te = ce[xyz];
\t\t{-d/(2a),-e/(2a)}
\t]

distance[l_List] := Module[
\t\t(* square of euclidean distance *)
\t\t{},
\t(l[[1,1]]-l[[2,1]])^2 +(l[[1,2]]-l[[2,2]])^2
\t\t\t]

allkvoronoi[points_List] := Module[
\t\t(* gets all voronoi circles and the points they contain *)
\t\t{l, rr, lc, circlelist, thiscircle, i, center, radius, otherpoints, k,h, \
j},
    
\t\tl = Length[points];
\t\trr = Range[l];
\t\t(* lc =  #of different circles *)
\t\tlc = Binomial[l,3];
\t\tcirclelist = {};
\t\tthiscircle = {1,2,3};
\t\t(* Use the undocumented function `NextKSubset' *)
\t\tFor[ i=1 , i <= lc,
\t\t\tcenter = circlecenter[ points[[thiscircle]] ];
\t\t\tradius = distance[{points[[thiscircle]][[1]], center}];
\t\t\totherpoints = Complement[ rr, thiscircle];
\t\t\tk = {};
\t\t\tFor[ j=1, j <= (l-3),
\t\t\t\th = otherpoints[[j]];
\t\t\t\tIf[ distance[{points[[h]],center}] < radius,
\t\t\t\t\t\tk=Append[k,h];
\t\t\t\t\t];
\t\t\t\tj++;
\t\t\t];
\t\t\tcirclelist = Append[circlelist, {i, thiscircle, N[center], k}];
\t\t\tthiscircle = NextKSubset[rr,thiscircle];
\t\t\ti++;
\t];
\tcirclelist
]\t

aroundvpoint[circle_,k_] := Module[
\t\t(* gets the edges and faces incident with a <
        k>-th order voronoi vertex (<circle>) *)
\t\t{nr,order, points, edges, nrinpoints, inpoints, regions},
\t\tnr = circle[[1]];
\t\torder = {1,3,2};
\t\torder2={3,1,2};
\t\tpoints = circle[[2]];
\t\tedges= KSubsets[ points,2];
\t\tinpoints = circle[[4]];
\t\tnrinpoints = Length[circle[[4]]];
\t\tIf[ nrinpoints== k-2,
\t\t\tregions = Table[Sort[Join[edges[[order2[[i]]]], inpoints]],{i,3}];
\t\t];
\t\tIf[ nrinpoints == k-1,
\t\t\tregions = Table[Sort[Append[inpoints, points[[order[[i]] ]] ] ],{i,3}];
\t\t];
\t\t{nr, edges, regions}
\t]

edgesandregions[struc_, k_] := Module[
\t\t(* Get all circles that contain k-2or k-1 points *)
\t\t{nr, kstruc,lci, i, ci},
\t\tnr = Length[struc];
\t\tkstruc = {};
\t  For[ i=1, i <= nr,
\t\t\tci = struc[[i]];
\t\t\tlci = Length[ci[[4]]];
\t\t\tIf[ lci== k-2  || lci == k-1,
\t\t\t\tkstruc = Append[kstruc, aroundvpoint[ci, k]];
\t\t\t];
\t\t\ti++;
  \t];
  \tkstruc
]

buildedges[ek_] := Module[
\t\t(* build all edges from the incident faces and edges info in <ek> *)
\t\t{cek, nrv, edges, i, edge, faces, ii, jj, al},
\t\tcek = ek;
\t\tnrv = Length[ cek ];
\t\tedges = {};
\t\tFor[ i=1, i < nrv,
\t\t\tFor[ j=1, j<= Length[ cek[[i,2]] ],
\t\t\t\tedge = cek[[i,2]][[j]];
\t\t\t\tfaces = cek[[i,3]];
\t\t\t\tFor[ ii=i+1, ii <= nrv,
\t\t\t\t\tal = Length[ cek[[ii,2]] ];
\t\t\t\t\tFor[ jj=1, jj <= al,
\t\t\t\t\t\tIf[ cek[[ii,2]][[ jj]] == edge,
\t\t\t\t\t\t\tIf[ Length[Intersection[ faces, cek[[ii,3]] ] ] == 2,
\t\t\t\t\t\t\t\tedges = Append[ edges, {cek[[i,1]], cek[[ii,1]] }];
\t\t\t\t\t\t\t\tcek = Delete[ cek, {ii,2,jj} ];
\t\t\t\t\t\t\t\tjj = 4;
\t\t\t\t\t\t\t  ii = nrv+1;
\t\t\t\t\t\t\t];
\t\t\t\t\t\t];
\t\t\t\t\t\tjj++;
\t\t\t\t\t];
\t\t\t\t\tii++;
\t\t\t\t];
\t\t\t j++;
\t\t];
\ti++;
];
edges
]\t\t
\t\t

showallkvoronoi[ points_] := Module[
\t\t(* Shows all n-1 Voronoi diagrams, defined by $n$ points,  simultanuously \
*)
      
\t\t{n, akv, alllines, k, edreg, edgerel, lines, i, b, e},
\t\tn = Length[ points ];
\t\takv = allkvoronoi[ points ];
\t\talllines = {Thickness[.01]};
\t\tFor[k=1, k <n,
\t\t\tedreg = edgesandregions[ akv, k];
\t\t\tedgerel = buildedges[edreg];
\t\t\tlines = {};
\t\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t\t];
\t\t\talllines  = Append[alllines, Join[ {Hue[N[k/n]]}, lines]];
\t\t\tk++;
\t\t];
\t\tShow[Graphics[alllines],PlotRange -> All, AspectRatio -> Automatic ,
      Axes -> False]
\t\t]\t\t

showallkvoronoi[ points_, range_] := Module[
\t\t(* Shows all n-1 Voronoi diagrams, defined by $n$ points,  simultanuously \
*)
      
\t\t{n, akv, alllines, k, edreg, edgerel, lines, i, b, e},
\t\tn = Length[ points ];
\t\takv = allkvoronoi[ points ];
\t\talllines = {Thickness[.004]};
\t\tFor[k=1, k <n,
\t\t\tedreg = edgesandregions[ akv, k];
\t\t\tedgerel = buildedges[edreg];
\t\t\tlines = {};
\t\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t\t];
\t\t\talllines  = Append[alllines, Join[ {Hue[N[k/n]]}, lines]];
\t\t\tk++;
\t\t];
\t\tShow[Graphics[alllines],PlotRange -> range, AspectRatio -> Automatic ,
      Axes -> True]
\t\t]\t\t

showkvoronoi[points_, k_Integer] := Module[
\t\t(* Shows the k-th order Voronoi diagram *)
\t\t{n, akv, alllines, edreg, edgerel, i, lines},
\t\tn = Length[ points ];
\t  akv = allkvoronoi[ points ];
\t  alllines = {Thickness[.002]};
\t\tedreg = edgesandregions[ akv, k];
\t\tedgerel = buildedges[edreg];
\t\tlines = {};
\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t];
\t\talllines = Join[{Thickness[.01], Hue[N[k/n]]}, lines];
\t\tShow[Graphics[alllines], PlotRange -> All,AspectRatio -> Automatic]
\t]\t

showsomekvoronoi[ points_,l_List] := Module[
\t\t(* Shows the combined Voronoi diagrams for the orders present in <l> *)
\t\t{n, akv, alllines, k, edreg, edgerel, lines, i, b, e,j},
\t\tn = Length[ points ];
\t\takv = allkvoronoi[ points ];
\t\talllines = {Thickness[.01]};
\t\tFor[j=1, j <=Length[l],
\t\t\tk = l[[j]];
\t\t\tedreg = edgesandregions[ akv, k];
\t\t\tedgerel = buildedges[edreg];
\t\t\tlines = {};
\t\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t\t];
\t\t\talllines  = Append[alllines, Join[ {Hue[N[k/n]]}, lines]];
\t\t\tj++;
\t\t];
\t\tShow[Graphics[alllines],PlotRange -> All, AspectRatio -> Automatic ,
      Axes -> True]
\t\t]\t\t

showsomekvoronoi[ points_,l_List, range_] := Module[
\t\t(* Shows the combined Voronoi diagrams for the orders present in <l> *)
\t\t{n, akv, alllines, k, edreg, edgerel, lines, i, b, e,j},
\t\tn = Length[ points ];
\t\takv = allkvoronoi[ points ];
\t\talllines = {Thickness[.01]};
\t\tFor[j=1, j <=Length[l],
\t\t\tk = l[[j]];
\t\t\tedreg = edgesandregions[ akv, k];
\t\t\tedgerel = buildedges[edreg];
\t\t\tlines = {};
\t\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t\t];
\t\t\talllines  = Append[alllines, Join[ {Hue[N[k/n]]}, lines]];
\t\t\tj++;
\t\t];
\t\tShow[Graphics[alllines],PlotRange -> range, AspectRatio -> Automatic ,
      Axes -> True]
\t\t]\t\t

voronoiflist[points_] := Module[
\t\t(* Returns all Voronoi faces defined by <points> *)
\t\t{n, akv, allfaces, i, faces, ear, j},
\t\tn = Length[points];
\t\takv = allkvoronoi[points];
\t\tallfaces = {{Range[n]}};
\t\tFor[ i=n-1, i >= 1,
\t\t\tfaces = {};
\t\t\tear = edgesandregions[akv,i];
\t\t\tFor[j=1, j <= Length[ear],
\t\t\t\tfaces = Join[faces, ear[[j,3]]];
\t\t\t\tj++;
\t\t\t];
\t\t\tallfaces = Append[allfaces, Union[faces]];
\t\t\ti--;
\t\t];
\t\tAppend[allfaces,{{}}]
\t\t]

randomnvor[n_Integer] := Module[
\t\t(* computes the euler chararacteristic for a set of <n> random points *)
\t\t{r, mp, flist, fvector, avector, euler},
\t\tr := Random[Integer,{0,1000}];
\t\tmp = Table[{r,r},{i,n}];
\t\tPrint[\"points = \",mp];
\t\tflist = voronoiflist[mp];
\t\tfvector=FVector[flist];
\t\tPrint[\"fvector = \", fvector];
\t\tavector=Table[(-1)^(i),{i,Length[fvector]}];
\t   euler = avector.fvector;
\t\tPrint[\"\[CurlyKappa] = \", euler];
\t\teuler
\t\t]

cvector[points_List] := Module[
\t\t(* Returns the number of circles of order $i$ for i=0,...,n-3 *)
\t\t{n, nr, cvec, akv, i},
\t\tn = Length[points];
\t\tnr = Binomial[n,3];
\t\tcvec = Table[0, {i, n-2}];
\t  akv = allkvoronoi[points];
\t\tFor[i=1, i <= nr,
\t\t\tcvec[[Length[ akv[[i,4]] ] + 1]]++;
\t\t\ti++;
\t\t];
\t\tcvec
]

 vvector[points_List] := Module[
\t\t(* Returns the number of vertices of the $k$-th order VD  for k=1,...,n-1 \
*)
      
\t\t{cvec,l,i},
\t\tcvec = Append[Prepend[ cvector[points],0],0];
\t\tl = Length[cvec];
\t\tTable[cvec[[i]] + cvec[[i+1]], {i,l-1}]
]

finfvector[points_] := Module[
\t\t{akv, finfvec, k, edreg, occur, vertices, lv, valencies, nrunbound, j},
\t\takv = allkvoronoi[ points ];
\t\tn= Length[points];
\t\tfinfvec = {};
\t\tFor[k=1, k <n,
\t\t\tedreg = edgesandregions[akv,k];
\t\t\toccur =Sort[Flatten[ buildedges[ edreg ] ] ];
\t\t\tvertices = Union[occur];
\t\t\tlv = Length[vertices];
\t\t\tvalencies = Table[Count[occur,vertices[[i]] ],{i,lv}];
\t\t\tnrunbound = 0;
\t\t\tFor[ j = 1, j <= lv,
\t\t\t\tnrunbound = nrunbound + (3-valencies[[j]]);
\t\t\t\tj++;
\t\t\t];
\t\t\t(* added later, RCL *)
\t\t\tIf[ lv == 0,
\t\t\t  nrunbound = 3;
\t\t\t];
\t\t\tfinfvec = Append[finfvec, nrunbound];
\t\t\tk++;
\t\t];
\t\tfinfvec
\t\t]\t\t

fvector[points_] := FVector[voronoiflist[points]]

evector[points_] := Module[
\t\t(* Returns the number of edges in all $k$-
        th order VD's  Note that the number of unbounded edges equals the 
          number of unbounded regions *)
\t\t{akv, finfvec, k, edreg, occur, vertices, lv, valencies, nrunbound, j},
\t\takv = allkvoronoi[ points ];
\t\tn= Length[points];
\t\tevec  = {};
\t\tfinfvec = {};
\t\tFor[k=1, k <n,
\t\t\tedreg = edgesandregions[akv,k];
\t\t\tbedges =  buildedges[ edreg ];
\t\t\toccur =Sort[Flatten[ bedges ] ];
\t\t\tvertices = Union[occur];
\t\t\tlv = Length[vertices];
\t\t\tvalencies = Table[Count[occur,vertices[[i]] ],{i,lv}];
\t\t\tnrunbound = 0;
\t\t\tFor[ j = 1, j <= lv,
\t\t\t\tnrunbound = nrunbound + (3-valencies[[j]]);
\t\t\t\tj++;
\t\t\t];
\t\t\t(* added later, RCL *)
\t\t\tIf[ lv == 0,
\t\t\t  nrunbound = 3;
\t\t\t];
\t\t\tfinfvec = Append[finfvec, nrunbound+Length[bedges]];
\t\t\tk++;
\t\t];
\t\tfinfvec
\t\t]\t\t


ffpoly[points_] := Module[
\t\t{rfvec, poly, i},
\t\trfvec = Reverse[fvector[points]];
\t\tpoly = 0;
\t\tFor[ i =1, i <= Length[rfvec],
\t\t\tpoly = poly + rfvec[[i]] t^(i-1);
\t\t\ti++;
\t\t];
\t\tFactor[poly]
\t]\t

chi[points_] := Module[
\t\t{flist, fvector, avector},
\t\tflist = voronoiflist[points];
\t\tfvector = FVector[flist];
\t\tavector= Table[(-1)^i,{i,Length[fvector]}];
\t\tavector.fvector+1
\t];

totalvertices[n_] := (1/3)n(n-1)(n-2)
totaledges[n_] := (1/2)n(n-1)^2
totalregions[n_] := (1/6)n(n^2+5)

fnk[n_,k_] := 2(k+1)(n-k)-(k+1)-(n-k)+2

bounds[vlist_] := Module[
\t\t{l,i,j},
\t\tl = Length[vlist];
\t\tTable[{Min[Table[vlist[[i,j]],{i,l}]],Max[Table[vlist[[i,j]],{i,l}]]},
    \t{j,Length[vlist[[1]]]}]
]

showsomecircles[points_List,k_Integer] := Module[
\t\t(* Shows the <points> and the Voronoi circles of order <k> *)
\t\t{n,akv,circles,i,center,radius},
\t\tn=Length[points];
\t\tIf[ k>n-3, Return[Print[\"k is too big...\"]] ];
\t\takv = allkvoronoi[points];
\t\tcircles = { Hue[N[k/n] ]};
\t\tFor[ i=1, i <= Length[akv],
\t\t\tIf[Length[ akv[[i,4]] ]==k,
\t\t\t\t  center =  akv[[i,3]];
\t\t\t\t  radius = distance[ {points[[  akv[[i,2,1]] ]], center }]
\t\t\t\t\t\t// Sqrt//N;
\t\t\t\t  circles = Append[circles,Circle[center,radius]];
\t\t\t];
\t\t\ti++;
\t\t];
\t\tl1 = Table[Text[i,points[[i]] ],{i,n}];
\t\t{Show[Graphics[{circles,l1}],
\t AspectRatio -> Automatic,
\t\t\tAxes -> False\t], Length[circles]-1}
]

<<$HOME/cas/matica/kvoronoi/ihull.m

bisector[lijst_] := Module[
\t\t\t{p1,a,b,p2,c,d},
\t\t
\t\t  p1 = lijst[[1]];
\t\t  a = p1[[1]]; b = p1[[2]];
\t\t  p2 = lijst[[2]];
\t\t  c = p2[[1]]; d = p2[[2]];
\t\t  If[d==b,1/2 (a +c),
\t\t  (a-c)/(d-b) x + (1/2) ( d+b +(c c-a a)/(d-b))]
\t\t\t]

bisector[p1_, p2_] := Module[
\t\t\t{a,b,c,d},
\t\t
\t\t  a = p1[[1]]; b = p1[[2]];
\t\t  c = p2[[1]]; d = p2[[2]];
\t\t  If[d==b,1/2 (a +c),
\t\t  (a-c)/(d-b) x + (1/2) ( d+b +(c c-a a)/(d-b))]
\t\t\t]


bisectorarr[points_] := Module[
\t\t{n,k,bis,x},
\t\tn = Length[points];
\t\tk = KSubsets[Range[n],2];
\t\tbis = Table[bisector[ points[[k[[i,1]] ]],points[[k[[i,2]] ]] ],
\t\t\t{i,Length[k]}]//N;
\t\tbis
\t\t]

pairtoposition[pair_,n_] := Module[
\t\t(* computes the position of an ordered pair in the enumeration of 
        combinations of length two of [n] *)
\t\t{p1,k},
\t\tp1=pair[[1]];
\t\tSum[n-k,{k,1,p1-1}]+ pair[[2]]-p1
\t\t]

oneslope[p1_,p2_] := Module[
\t\t\t{a,b,c,d},
\t\t  a = p1[[1]]; b = p1[[2]];
\t\t  c = p2[[1]]; d = p2[[2]];
\t\t  If[d==b,1/2 (a +c)//N, (a-c)/(d-b) //N]
\t\t\t]

allslopes[points_] := Module[
\t\t{n,k,i},
\t\tn = Length[points];
\t\tk = KSubsets[Range[n],2];
\t Table[oneslope[ points[[ k[[i,1]] ]],points[[ k[[i,2]] ]] ],
\t\t\t{i,Length[k]} ]
\t\t]

buildedgesinf[ek_] := Module[
\t\t(* build all edges from the incident faces and edges info in <ek> *)
\t\t{cek, nrv, edges, i,j, edge, faces, ii, jj, al},
\t\tcek = ek;
\t\tnrv = Length[ cek ];
\t\tedges = {};
\t\tinfedges={};
\t\tFor[ i=1, i < nrv,
\t\t\tFor[ j=1, j<= Length[ cek[[i,2]] ],
\t\t\t\tedge = cek[[i,2]][[j]];
\t\t\t\tfaces = cek[[i,3]];
\t\t\t\tFor[ ii=i+1, ii <= nrv,
\t\t\t\t\tal = Length[ cek[[ii,2]] ];
\t\t\t\t\tFor[ jj=1, jj <= al,
\t\t\t\t\t\tIf[ cek[[ii,2]][[ jj]] == edge,
\t\t\t\t\t\t\tIf[ Length[Intersection[ faces, cek[[ii,3]] ] ] == 2,
\t\t\t\t\t\t\t\tedges = Append[ edges, {cek[[i,1]], cek[[ii,1]] }];
\t\t\t\t\t\t\t\tcek = Delete[ cek, {ii,2,jj} ];
\t\t\t\t\t\t\t\tjj = 5;
\t\t\t\t\t\t\t  ii = nrv+1;
\t\t\t\t\t\t\t];
\t\t\t\t\t\t];
\t\t\t\t\t\tjj++;
\t\t\t\t\t];
\t\t\t\t\tii++;
\t\t\t\t];
\t\t\t\tIf[ jj < 5, infedges=Append[infedges,{i,edge}]];
\t\t\t j++;
\t\t];
\ti++;
];
For[k=1, k <= Length[cek[[nrv,2]]],
\t\t\tinfedges = Append[infedges, {i,cek[[nrv,2,k]]} ];
\t\t\tk++;
];
{edges,infedges}
]\t\t

sign[p_,q_,r_] := Module[
\t\t{},
     Det[{{1,p[[1]],p[[2]]},
\t\t\t\t\t{1,q[[1]],q[[2]]},
\t\t\t\t\t{1,r[[1]],r[[2]]}}]//Sign
\t\t]

showkvoronoiinf[points_, k_Integer] := Module[
\t\t{lines,n, akv,edreg, edgeinfo,edgerel,i,b, e,infinfo, infedge,thiscircle,
      slopes, slope, centre,radius,ss,s1,p1label, p2label, p3label,  p1, p2, 
      p3,alllines, tpoints, cyoung, testsign},
                If[ k < 1, Return[{}]];
\t\tlines = {};
\t\tn = Length[ points ];
\t\tIf[ k >= n, Return[{}]];
\t  akv = allkvoronoi[ points ];
\t\tedreg = edgesandregions[ akv, k];
\t\tedgeinfo = buildedgesinf[edreg];
\t\tedgerel = edgeinfo[[1]];
\t\t(* Add the edges *)
\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t];
\t\t(* Now we add the halflines *)
\t   slopes = allslopes[points];
\t\tinfinfo = edgeinfo[[2]];
\t\tFor[i=1, i <= Length[infinfo],
\t\t\tinfedge = infinfo[[i]];
\t\t\tthiscircle=edreg[[infedge[[1]],1]] ;
\t\t\t(* determine whether <thiscircle> is young or old *)
\t  \thcircle = Length[ akv[[thiscircle,4]] ];
\t\t\tIf[ hcircle+2==k, cyoung = -1, cyoung = 1];
\t\t\tslope= slopes[[pairtoposition[infedge[[2]],n ] ]];
\t\t\tcentre= akv[[ edreg[[infedge[[1]],1]],3]];
\t\t\tradius = N[Sqrt[ distance[ {centre, points[[infedge[[2,1]] ]]} ] ] ];
\t\t\tss = (2.radius)/(Sqrt[1+slope^2]){1,slope};
\t\t\ts1 = ss+centre;
\t\t\tp1label=infedge[[2,1]];
\t\t\tp2label=infedge[[2,2]];
\t\t\tp3label= Complement[akv[[thiscircle,2]],{p1label,p2label}][[1]];
\t\t\tp1 = points[[p1label]];
\t\t\tp2 = points[[p2label]];
\t\t\tp3= points[[p3label]];
\t\t\ttestsign = sign[p1,p2, s1]  sign[ p1, p2, p3 ];
\t\t\tIf[ cyoung== testsign,
\t\t\t\t\t\t\t\t\ts1 = -ss + centre;
\t\t\t];
\t\t\tlines = Append[lines,Line[{centre,s1}]];
\t\t\ti++;
\t\t];
\t\talllines = Join[{Thickness[.005], Hue[N[k/n]]}, lines];
\t\ttpoints = Table[Text[i,points[[i]] ],{i,n}];
     Show[Graphics[{alllines,tpoints}], PlotRange -> All,
      AspectRatio -> Automatic]
\t]\t

showkvoronoiinf[points_, k_Integer,akv_ ]:= Module[
\t\t{lines,n, akv,edreg, edgeinfo,edgerel,i,b, e,infinfo, infedge,thiscircle,
      slopes, slope, centre,radius,ss,s1,p1label, p2label, p3label,  p1, p2, 
      p3,alllines, tpoints, cyoung, testsign},
\t\tlines = {};
\t\tn = Length[ points ];
\t\tIf[ k >= n, Return[{}]];
\t\tedreg = edgesandregions[ akv, k];
\t\tedgeinfo = buildedgesinf[edreg];
\t\tedgerel = edgeinfo[[1]];
\t\t(* Add the edges *)
\t\tFor[ i = 1, i <= Length[ edgerel],
\t\t\t\tb  = edgerel[[i,1]];
\t\t\t\te = edgerel[[i,2]];
\t\t\t\tlines = Append[ lines,Line[ {akv[[b,3]], akv[[e,3]]}]];
\t\t\t  i++;
\t\t];
\t\t(* Now we add the halflines *)
\t   slopes = allslopes[points];
\t\tinfinfo = edgeinfo[[2]];
\t\tFor[i=1, i <= Length[infinfo],
\t\t\tinfedge = infinfo[[i]];
\t\t\tthiscircle=edreg[[infedge[[1]],1]] ;
\t\t\t(* determine whether <thiscircle> is young or old *)
\t  \thcircle = Length[ akv[[thiscircle,4]] ];
\t\t\tIf[ hcircle+2==k, cyoung = -1, cyoung = 1];
\t\t\tslope= slopes[[pairtoposition[infedge[[2]],n ] ]];
\t\t\tcentre= akv[[ edreg[[infedge[[1]],1]],3]];
\t\t\tradius = N[Sqrt[ distance[ {centre, points[[infedge[[2,1]] ]]} ] ] ];
\t\t\tss = (2.radius)/(Sqrt[1+slope^2]){1,slope};
\t\t\ts1 = ss+centre;
\t\t\tp1label=infedge[[2,1]];
\t\t\tp2label=infedge[[2,2]];
\t\t\tp3label= Complement[akv[[thiscircle,2]],{p1label,p2label}][[1]];
\t\t\tp1 = points[[p1label]];
\t\t\tp2 = points[[p2label]];
\t\t\tp3= points[[p3label]];
\t\t\ttestsign = sign[p1,p2, s1]  sign[ p1, p2, p3 ];
\t\t\tIf[ cyoung== testsign,
\t\t\t\t\t\t\t\t\ts1 = -ss + centre;
\t\t\t];
\t\t\tlines = Append[lines,Line[{centre,s1}]];
\t\t\ti++;
\t\t];
\t\talllines = Join[{Thickness[.005], Hue[N[k/n]]}, lines];
\t\ttpoints = Table[Text[i,points[[i]] ],{i,n}];
     Show[Graphics[{alllines,tpoints}], PlotRange -> All,
      AspectRatio -> Automatic]
\t]\t

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Now suppose we want to create the k-th Voronoi diagram. The \
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incident edges and faces. 

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Using the edges and regions incident with a Voronoi vertex at level \
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\>", "Subsection",
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When we know all edge relations, we can build Voronoi diagrams for \
every order at the same time or apart\
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We can also compute a list of all faces present in all diagrams, \
look at the f-vector, check the Euler-Poincare formula and draw the \
poset.\
\>", "Subsection"],

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