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[David Canino <>]

Dear Pranav,

the terms "manifold" and "quasi manifold" are not typical of CGAL, to the best of my experience. They are classes of complexes with particular properties in topology. I spent 3 years of my PhD in Computer Science on this. You can find their definitions in any reference of topology. For instance, you can refer this article, "A Compact Representation for Topological Decompositions of Non-Manifold Shapes", David Canino, Leila De Floriani, GRAPP 2013, available on the web, just to mention one. Here, some relations with other type of complexes are also highlighted. In the above paper, you can also find notions regarding manifold/non-manifold. Specifically, a non-manifold point p has a neighborhood such that it cannot be deformed into the triangulation of the sphere, centered on p, without creating new holes. This means that the neighborhood of p is mostly disconnected (but not necessarily). For example, if you consider a torus, you can always do this for each point. Any continuous deformation of torus also satisfies this property. If you consider two triangles sharing only a vertex v, then the neighborhood of v cannot be mapped on a sphere, since it results disconnected. Specifically, it is contracted at point v, which is a non-manifold singularity. A more convenient way to see this consists of considering the link of any vertex, since it results disconnected in two components, in this case.

Best regards

David Canino

2014-06-09 14:46 GMT+02:00 Pranav <>:

I want to understand difference between Manifolds and quasi-manifolds.
Quoting a para in  section
<http://doc.cgal.org/latest/Combinatorial_map/index.html> *2.4 Combinatorial
Map Properties*

"...In 2D, quasi-manifolds are manifolds, but this is no longer true in
higher dimension as we can see in the example presented in Figure 24.6. In
this example, the object to the right is not a manifold since the
neighborhood of the point p in the object is not homeomorphic to a 3D ball
(intuitively, two objects are homeomorphic if each object can be
continuously deformed into the second one; in such a case, the two objects
have exactly the same topological properties)..."

Can anyone explain what does it mean(geometrically) to say : *"...the
neighborhood of the point p in the object is not homeomorphic to a 3D
ball..."*?

Is 3D-ball same as  3-sphere <http://en.wikipedia.org/wiki/3-sphere>  ?

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