CGAL users discussion list

["Wesley Smith" <>]

I realize that is the case in general, but for the dual of a vertex on
the convex hull, isn't it just going to be a series of voronoi facets
that arc in and out of the convex hull defined by a series of edges
like ray->segment->...->segment->ray where there can be 0 to N
segments between the rays?

wes

On 9/24/07, Andreas Fabri 
<>
 wrote:
> Wesley Smith wrote:
> > Here's a tricky problem I've been wrestling with that I'm wondering if
> > anyone has any insight into.
> >
> > If a vertex is part of the convex hull of a 3D delaunay triangulation,
> > then in the set of adjacent facets forms a conical structure which can
> > be iterated over to derive a circulator over the corresponding voronoi
> > edges with the addition of a rule that says if there is a facet not on
> > the crust between 2 such facets, it must be visited before moving to
> > the adjacent crust facet.
>
> Hi Wes,
>
> I am not sure to understand the order you impose. There can be an
> arbitrary number of Voronoi facets which do not intersect the
> convex hull, and they are not necessarily "between" facets that
> intersect the convex hull.
>
>
> andreas
>
> >
> > This is what I've implemented so far.  Now the trick is to extend this
> > to delaunay vertices internal to the triangulation.  From what I can
> > see, the adjacent facets to such a vertex form a number of closed
> > loops which need to be iterated over.  Now my question is, is there a
> > geometric property that can be exploited to uniquely iterator over all
> > such facets such that each successive facet in the iteration is
> > adjacent to the previous using only local information (i.e. I'm at
> > facet f and with vertex v as my pivot, what is the "next" facet)?
> >
> > For instance, imagine a 3x3x3 grid.  The dual of the point at the
> > center of the grid is a cube.  Thus there are 6 loops to visit.  How
> > does one move between the loops with only local information?  Or, is
> > this not possible and I need to keep track of when a loop is closed
> > and move on to the next one?
> >
> > thanks,
> > wes
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