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The observer is used to construct the first 2 arrangements.
Then, while doing the overlay, you need to use the overlay traits.
It's all in the manual.

Quoting Graviton 
<>:

> Hello,
>
> I can't quite see how the observer can keep track of the original face, as
> when the time when I input the arrangement, I don't input in the original
> face information. Mind to be more explicit about it?
>
> On Tue, Nov 8, 2011 at 6:31 PM, Efraim Fogel [via cgal-discuss] <
> >
>  wrote:
>
> > Extend the cells of the DCEL of the two arrangements [1] and use an
> > observer [2] to update the additional fields when the arrangements are
> > constructed. Then, use the overlay traits [3] to update the final
> > arrangement while performing the overlay operation.
> >
> > [1]
> >
> > http://www.cgal.org/Manual/latest/doc_html/cgal_manual/Arrangement_on_surface_2/Chapter_main.html#Section_31.8
> > [2]
> >
> > http://www.cgal.org/Manual/latest/doc_html/cgal_manual/Arrangement_on_surface_2/Chapter_main.html#Section_31.7
> > [3]
> >
> > http://www.cgal.org/Manual/latest/doc_html/cgal_manual/Arrangement_on_surface_2/Chapter_main.html#Section_31.9
> >
> > Graviton wrote:
> >
> > > I have two  http://en.wikipedia.org/wiki/Polygon_mesh  polygonal mesh
> >  , with
> > > all the usual properties such as vertex and edge and the faces.
> > >
> > > Now I want to conduct an overlay between the two, so what I do is, I
> > create
> > > two arrangements, each for one polygonal mesh, and then use the overlay
> > > method to come up with the resultant.
> > >
> > > The problem is, how to construct each arrangement, such that for each
> > face
> > > in the arrangement, I can trace it back to the original face in the
> > > polygonal mesh?  For each face in the polygonal mesh, there is a unique
> > > meaning assigned to it, so the arrangement has to know this when it is
> > > constructed.
> > >
> > >
> > >
> > > --
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