Higher order Voronoi diagram(Open 5/Sep/2000 : The 5th Revision Thursday, 03-Jun-2010 22:12:25 JST)

If we use Higher order Voronoi diagram, we can find out the 2nd or 3rd nearest facility.

White is the first order, green is 2nd, and blue is 3rd.

Algorithm for the higher order Voronoi diagrams (If you want draw the l-th order Voronoi diagram, then m=1,2,...,l)

i=1,...,N-1 j=i+1,...,N Consider a bisector of p(i) and p(j) k=1,...,N except for i and j Consider a bisector of p(i) and p(k) Calculate points of intersection of bisector(i,j) and bisector(i,k) next k Add the points at x=0 and x=(the width of the screen) of the bisector of i and j into points of intersections Sort points of intersections in turns of x coordinates k=1,...,the number of interval of points of intersections Let c be a middle point of interval of points of intersection Let d be d(c,p(i)) label=0 h=1,...,N except for i and j Let d' be d(c,p(h)) If d'<d then label=label+1 next h If label=m-1, then draw the interval of points of intersection by m-th order's color. next k next j next iAnd the following is the algorithm of the ordinary Voronoi diagram. It is very similar.

i=1,...,N-1 j=i+1,...,N Consider a bisector of p(i) and p(j) k=1,...,N except for i and j Consider a bisector of p(i) and p(k) Calculate points of intersection of bisector(i,j) and bisector(i,k) next k Add the points at x=0 and x=(the width of the screen) of the bisector of i and j into points of intersections Sort points of intersections in turns of x coordinates k=1,...,the number of interval of points of intersections Let c be a middle point of interval of points of intersection Let d be d(c,p(i)) label=0 h=1,...,N except for i and j Let d' be d(c,p(h)) If d'<d then label=1 next h If label=0, then draw the interval of points of intersection next k next j next iJava(hivoro.java)

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