CGAL users discussion list

[Jeffrey Bush <>]

My "working definition" mainly applied to manifolds and I tried to update it, unsuccessfully, to quasi-manifolds. Sorry for adding to the confusion. I stopped using the LCC when it didn't suit my needs (I needed to be able to have things like figure 24.7a) so I never developed a "working definition" for quasi-manifold.

For quasi-manifolds it seems that as long as every n-cell only touches other n-cells and they only touch at shared (n-1)-cells you are good. So a 3-cell must touch other 3-cells with a shared 2-cell (facet), the two parts of the shared 2-cell are touching each other at 1-cells (edges), and those 1-cells are touching at 0-cells. You cannot skip dimensions (so two 3-cells cannot touch at 1-cells (edges) without touching at 2-cells as well).

The reason the figure you point to is a quasi-manifold is that the 0-cell "p" is shared through the shared 1-cells (edges) and shared 2-cells (facets). Thinking in reverse, the four 3-cells touch each other at 2-cells, all 2-cells that touch touch at 1-cells, and all the 1-cells that touch touch at 0-cells. You never skip a dimension. If there was any space between the four 3-cells (or if you were missing two opposite 3-cells) and the 0-cell p was still shared then it would no longer be quasi-manifold. It does not matter that two 3-cells only touch each other at a single 0-cell as long as you can use a chain of touching 2-cells and 3-cells to reach the other 3-cell.

Since I am not a geometer in any sense, please take my responses with a grain of salt. For robust answers you will have to go with the definitions others have given, but to me they are difficult to understand since the only geometry course I have taken was in the beginning of high school. The definition I gave you for manifold is from 3D modelling circles and I have found it works very well for what I need and is fairly easy to understand for the non-geometer.

Jeff

On Mon, Jun 16, 2014 at 3:56 AM, Guillaume Damiand <> wrote:

Le 16/06/2014 09:38, Pranav a écrit :

Could you justify how  this figure
<http://doc.cgal.org/latest/Combinatorial_map/index.html#fig__figquasivariete>
(object with vertex /p/) represents a quasi-manifold object?

This is a 3D quasi-manifold because it is a set of 2D quasi manifolds that are glued along faces.

I agree that one can cover all faces and edges of the object incident at
/p/, but does it make it quasi-manifold even if opposite pentagons are
/directly/ connected at a vertex(i.e., /p/) only.

I am a bit confused here.



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