Inside, outside.(Viewpoint MODELING)

When building a virtual model of the real world for computer graphics applications, we usually represent solid objects as polygonal, NURBS, or other types of surfaces. This makes the representation, the modeling, and the rendering of objects much easier. Yet, there are some interesting questions regarding surfaces when representing volumes: Are closed and hole-less surfaces always defining 3D volumes? Do all surfaces have two sides? Why does our favorite modeling application fail at computing the union or the intersection of certain objects? And what is the mysterious error message regarding manifolds?

Rigid objects, no matter how thin, always take up some volume in 3D space. It is never a good idea to ignore the thickness of even the finest sheets or plates, especially if they are made of plastic or metal. Our eyes always pick up the highlights on the sharp but still round edges--an essential visual sign of thinness. Modeling polygon meshes that do not perfectly encompass a 3D volume may also cause headaches at different stages of the pipeline: Automatic polygon reduction or collision-detection algorithms likely will fail on our model, we won't be able to use Boolean operations on the object, and rendering artifacts may appear on the final images.

[ILLUSTRATION OMITTED]

Manifolds

In order to understand the reason why these problems arise, and to be able to fix them, we should look at some results of the theory of geometric topology. The first concept one should understand here is the manifold. Instead of shocking you with some pure mathematical definitions, let's look at the simplest example: the one-dimensional manifold. In a one-manifold, every point has a neighborhood that looks like a line segment. Thus, a one-dimensional manifold is a curved or straight line, with no discontinuities or junctions. Manifolds also can be open (like a curved line) or closed (like a circle).

[ILLUSTRATION OMITTED]

Two-dimensional manifolds--referring to surfaces in 3D space have a similar definition: In a two-manifold, every point has a neighborhood that looks like a continuous disk. That is, the surface can be locally deformed into a plane, without tearing it or identifying separate points. An example of a closed, two-dimensional manifold would be a sphere, torus, or the Klein bottle. A valid two-manifold topology polygon mesh never has edges shared by more than two faces (that would be similar to a "T" junction). Essentially, two-manifolds are made of a single sheet of totally flexible, imaginary fabric. Manifolds are defined by a local constraint on the surface. That is good news, as it is very easy to check a polygon mesh by simply inspecting every vertex and edge for problematic topological configurations.