A good 3D topology has to have consistent parallel lines. Once it is smoothed, there should be the minimum amount of irregular vertexes.
A normal/good vertex consists of 4 points, bad or irregular singularities consist of 3 or 5 vertex points.
Bad singularities will cause problems once the object is rendered. This is due to the fact that the mesh is not connected in a regular way which will cause irregular movement once light is applied. The 3 or 5 point edges/vertexes will show flaws when a light or a shadow is cast on it because it will be connected in an unusual way compared to the rest of the mesh.
A good mesh will consist of parallel lines through out and will have mainly 4 point vertices. 4 point vertices allow smooth movement and rendering due to the fact that each and every one is consistent and its a lot easier to render it because the same algorithm for rendering can be applied throughout with no interruptions in the meshes flow.
Once you get to a 3 point or a 5 point vertex the algorithm is disturbed due to the fact that a normal amount of polys has been disturbed and either extra has been added or some polys are missing.
This is an example of a very good topology:
If you look at this mesh, you will notice that all of the lines are consistently parallel and there is a very minimum amount of 3 or 5 point irregularities.
In this picture the only bad thing is that there is way too many polys in the mouth region, this will make the rendering process very difficult and time consuming. If any errors were to arise, fixing errors would be tedious and very technical due to the amount of detail in the polys.
This is another good example of a topology due to the very clear and minimum amount of polys:
And this is an example of a good but flawed topology:
By looking at this topology you can notice right away at least 2, 5 point vertex points. This is without actually zooming in and examining the mesh fully. The first 5 point vertex is on the characters cheek, this will cause problems when the character is rendered because this point is in a very visible are (right on the cheek). This may look good now while it is in a wire mesh, but once you throw smoothing on it in Maya and add some lighting to that specific area you will see polygon inconsistencies due to the out of character point junctions.
This is an example of very bad practice when it comes to using vertex points and keeping the grid as constant as possible. This is a character off of the Lord of the Rings films. It is made to look good and very suffisticated but the amount of work that had to go into hiding the imperfections of the finished rendered product cost the developers/the movie investor a load of extra money. This is due to the fact that the 3d modelers and animators had to go through the task of adding a load of extra layers over just the face to cover up all of the imperfections.
If the lines that are highlighted are examined more closely, the blatant amount of 5 and 3 point singularities is immense. This is due to the fact that either A. the 3d modeler didn’t plan out their mesh properly or stick to the strict guidlines which help the rest of the team in the post production stages, or B. The 3d modeler took the detail purposely over the top to try and make it seem more realistic and lifelike.
Both of these points have their pluses and minuses as they can bot be justified in different ways, realism being one of them and orthodox planning being the other for originality and one of a kind outcome.
The thing is though, a 3d mesh does not necessarily have to have immense amounts of polygons and detail to make it look great. There is speculation that the same or very near like replicas could have and can be made with using half the polygons. Which would eliminate a load of the 3 and 5 star singularity points. The whole end product quality lies in the structure and planning of the 3d mesh. There are a load of surfaces, uv maps, textures and effects that can be put on in the post production stages to boost the quality of the original product by vastly marginal quality manipulations. There are smoothing tools that can be applied at some cost of rendering time and collision detection areas but the final product would be as good as the mesh used above minus the tedious work that had to be put into hiding the irregularities. Hence saving time, money and resources to reach a desired look and feel of the final product. In the end it all falls down to good practice, planning and understanding of all the processes that are associated with creating such models and animations.
P.I.G 