A surface is a hard thing to describe, sometimes! A polygon model can be considered a surface model, or a collection of surfaces. But some polygon models represent the boundary of manifold solid bodies. Its kind of context sensitive. I think you have a good start of an understanding in S(u,v) -> (x,y,z). To generalize that, a surface is a locus of points that can somehow be represented by a function of exactly two indepedent parameters, with the function being at least G0 continuous within the boundaries. The G0 continuity means that the surface must be geometrically continuous without gaps, but all derivatives can be discontinous---allowing a polygon model to be defined as a surface. That is sort of close to a mathematical definition.

There are some more concise, but similar definitions over at Dictionary.com

Quote:Original post by Nene
Therefore, a polygonal model is not always a surface model, is it?

For example, a face will hardly be represented as a surface, as it probably will contain gaps, discontinuities and irregular boundaries. (It could be modelled using a set of surfaces, of course)

I'll ramble for a bit in a minute, but first let me say that when you call something a "surface model," it could simply mean a model made from one or more surfaces. So, from a semantics point-of-view, a polygon model could always be considered a surface model.

It can be fuzzy whether you consider it to be one surface or multiple surfaces. Now, the ramble.

Well, irregular boundaries are allowed. And holes are allowed. So a face could be done with a single surface (by your current understanding).

But, er....there are subtleties... My statement of G0 continuity should have been qualified. I should have included the word locally in many of my sentences.

I didn't mean to say that the parameterization coordinates, (u,v) had to map to a globally continuous part of 3D space. Non-degenerate regions in (u,v) space only have to map to a locally continuous part of 3D space. Different non-degenerate regions in (u,v) space can map to entirely disconnected parts of 3D space. Think texture mapping with the unwrap feature of 3D modeling packages. When you texture map, say, a vehicle, you'll often have a single texture map, but different parts of the vehicle map to different parts of the texture. One corner of (u,v) space might map to the door, but another part to the dashboard control panel. You might consider the door and the dashboard to be different surfaces, but from they could be could, strictly speaking, treated/considered as the same surface. They can be added together and still be considered a single 2D subspace within a 3D space. That allowance would certainly break the assumptions/conditions of some theorems of calculus, e.g., certain theorems wouldn't apply across local boundaries. But it is the point-of-view that I take in our computational geometry engine at work. And, it makes for some very nice, extremely elegant implementations of computational geometry (including Boolean operations, surface simplification, other things).

I'll try to give a straightforward example of the idea of locality. When doing, for example, Boolean or set operations, you could consider a generalized surface to be one that has multiple regions that are not connected. There is in computer-aided design the concept of trimmed surfaces, such as trimmed NURBS. Here, you have one nice surface that has your typical S(u,v)->(x,y,z) mapping. Then, you break it up into pieces by applying rules to trim it. You still have the same underlying surface, but with disconnected bits. The solid modeling package, ACIS calls these disconnected bits "lumps"---their terminology. Take, for example, a cylindrical surface. That's your base surface. Then take two spheres that both intersect the surface but not each other. Trim the cylindrical surfaces so that you only keep what is inside the two spheres. Now you have two discreet parts of the same surface. Locally the surfaces are at least G0 continuous, and you can use a simple S(u,v)->(x,y,z) mapping within the boundaries of each discreet part.

So, the real thing I'm getting at here is that most of what I said before technically needs to have the word locally applied to it. Although if you're more comfortable thinking about it globally, that is okay too, :).

Quote:Original post by NeneBun then, in this case we must talk about polygonal simplification instead of surface simplification.

Is this right?¿ Or do we could continue speaking about surface simplification although we had a irregular polygonal model?

In general, I'd still call it surface simplification. If you use the quadric error metric approach to simplifying surface models that happen to be constructed of polygons, you will find that you can add constraints at the boundaries that will properly retain boundaries in the case that your model has many groups (lumps?) of polygons, the polygons within each group connected together----possibly with holes in a group and a group having an irregular boundary, but the separate groups not touching each other. That approach to simplification will generate a new model---still based on polygons but also still considered a surface model. (The quadric error metric approach does require that the surfaces be manifold, though----every edge must touch no more than 2 subfaces/triangles).