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THANK YOU VERY MUCH!
I begin to understand the concept now and i'm impressed. Only one point is
left. I read in my 3d points with:

ifstream in(surface.dat);
istream_iterator<Point> begin(in);
istream_iterator<Point> end;
Delaunay2.insert(begin, end);

I do the triangulation and assign indices to the vertices. After that i want 
to
store the 3d points in a new file so that i can work with the new order. When 
i
try to get out the coordinates from the vertices i have a problem.

fvii->info()= index++;
Point p = fvii->point();
cout << p.hx() << " " << p.hy() << " " << p.hz()<< endl;

//with
//struct K : public CGAL::Exact_predicates_inexact_constructions_kernel {};
//typedef K::Point_3   Point;

The input file has the format:

123456.789 1234567.890 123.456

However if i write the coordinates to an output file i get the following
format:

123456 1.23456e+06 123.45

Does the Kernel uses double? I also tried p.cartesian() and p.homogeneous()