CGAL users discussion list

[Jeffrey Bush <>]

Like myself, you don't seem to be a geometer. From the non-geometer standpoint, my working definition of a manifold is:

• Every edge belongs to two facets (except border edges, if allowed, which only belong to one facet)
• Every vertex is surrounded by one sequence of edges and facets*
• All normals face inside or outside, but not both
• Facets only intersect each other at common edges/vertices (and not, for example, having two facets intersect each other in the middle of each other)
  

The first two are checked with Polyhedron_3::is_valid() [and should always be true unless you built the polyhedron with the low-level functions], the third cannot be violated with Polyhedron_3 at all, and there is no easy way to check the last one (people have suggested converting to a Nef Polyhedron and then using is_valid() which should check this).

* The second bullet point is probably the most confusing. An easy way (but possibly incorrect) to think about this is that two facets cannot touch at a single vertex only. An example of a bad situation of this is if you took an hourglass shape and pinched the middle down to a single vertex. You would no longer be able to reach all of the incident edges of that vertex by starting with one incident edge and going around to the other incident edges connected by a facet.

A quasi-manifold relaxes the above definition slightly, mainly with the first bullet point. Edges are now allowed to belong to one or more facets. Because of this the third bullet point also is different, but shouldn't really matter since it is forced on you by the implementation. Importantly, the second bullet point is still true (basically two facets cannot touch just at a single vertex). Once again, the last bullet-point is not enforced by the implementation and is up to you to check.

Jeff

On Thu, Jun 12, 2014 at 12:08 AM, Guillaume Damiand <> wrote:

Le 11/06/2014 13:32, Pranav a écrit :

Actually, I want to get intuition behind homeomorphism. For example in
Figure 24.6
<http://doc.cgal.org/latest/Combinatorial_map/index.html#fig__figquasivariete>
, how can I /intuitvely/ figure out that the neighbourhood of /p/ cannot be
deformed to 3-ball?

Once again, cf wikipedia [1]: "Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape."

[1] http://en.wikipedia.org/wiki/Homeomorphism

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