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Manifold vs Non-Manifold Geometry

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I am not able to figure out the difference between the manifold and non-manifold geometry.

From literature, I understand that geometries such as Sphere,Ellipse,Box etc are examples of manifold geometry. and examples of non-manifold geometry, includes all self interesecting objects, objects with hole.

But I want to learn the key mathematical properties, which differentiate manifold from non-manifold geometry? In particular, I need to understand, how in an application, we recongnize whether the given geometry is of manifold or non-manifold kind.

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A nice introduction to the topic would the Donal O'Shea's "The Poincaré Conjecture".
For mathematical deepness, there would be "Differential Geometry" by Erwin Kreyszig.

And afaik there is no non-manifold geometry (see also this for a quick intro), and afaik2, topology is exactly the discipline of classifying things as number of holes, loops, and some more. E.g., think of the Moebius strip, which is a famous example of topology, and, it has a hole.

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I studied manifolds in my former life as a mathematician more deeply than I ever wanted to.

The very basic idea is that something is a manifold if around each point there is a little piece that looks like a copy of Rn. Things around a self-intersection point don't look like Rn at all, so anything containing a self-intersection point is not a manifold. Depending on what you mean by "hole", there are perfectly good manifolds with holes, like the circle.

The word "manifold" needs an adjective to go with it. The most common are algebraic manifolds, analytic manifolds, Cinfinity manifolds, Ck manifolds (for each natural number k) and topological manifolds. Very often the context tells you what type of manifolds you are interested in, and then the adjective might be dropped.

For a set M to be a manifold we need to have coordinates around each point, which are applicable in an open set around the point. These coordinates establish a bijection from an open set in Rn to a subset of M, and are called charts. If the subsets of M mapped by two charts have an intersection, we can consider the function that would map the coordinates in one chart to the coordinates in the other chart, so it is a function from some open set of R^n to another open set of R^n. If these change-of-chart mappings are all continuous and they have two continuous derivatives, we say this is a C2 manifold. Similarly for the other types of manifolds. The type of manifold you are dealing with is determined by the smoothness of the change-of-chart functions.

A lot of geometry only deals with manifolds, but sometimes a larger class of objects is considered. For instance, one can investigate manifolds with border or manifolds with singularities.

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Here's a good example of what the difference is:
http://www.transmagic.com/node/118

Quote:

In other words manifold means: You could machine the shape out of a single block of metal....and with a non-manifold shape you could not.


In the case of the two cubes that they show (a closed non-manifold mesh), you can see that the edge where the two cubes touch is shared by 4 faces (the edge belongs to two faces per cube).

With a closed manifold mesh (which is pretty much the most common type of mesh in gamedev), you would find that any edge will only belong to two faces. In an open manifold mesh (like a closed manifold mesh that has been chopped in half basically), there will be a boundary, and so some edges might belong to only one face.

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