Here are some of my own illustrations to show my understanding of anatomy and further understanding of facial topology

Poles are handy when you are trying to extrude something, as I described earlier with rears and eyes. But when it comes to loops which define the flow of the topology it is best keep to quads and 4 edge axis, this gives a more organic and natural look. Poles should be used when these loops interact. When loops are following the same path and then split to look round different parts of the topology, at that split is where you will find E or N poles. See just below the eye, above and below the mouth and below the ear. In fact I guess if you could work and and map where all the poles are the rest of the four point flowing topology would just fall into place.

Further study into the mathematical side of topology shows me that the ideal polygon mapping is in quads not triangles. Triangles can produce un even topology wheres as quads will always produce symmetrical typology, with an even spread. See image above.

MUSCULAR UNDERSTANDING

http://www-2.cs.cmu.edu/afs/cs/project/face/www/facs.htm When studying this site, it showed me many different facial movements and which muscles are employed to execute them. When referring to the earlier image I posted of the muscular structure of the human face I can grasp a far greater understanding of facial movement. After this understanding, if I look at a the typical topological layout of a 3D rig, Like the one above, I can instantly understand WHY it has been constructed that way and not just how it works.

DEJA VU!

From studying this thread at http://www.subdivisionmodeling.com/forums/showthread.php?t=8000 I have been able to get more of an undestanding of the more complicated mathematical side if topology. Looking at "E" and "N" poles. An "E" pole has 5 edges connected to it and an "N" pole has 3. These are the ideal poles when making making a curve in the shape or an extrusion such as the eyes or the ears and you need to make a break in the basic grid layout. This makes perfect sense when looking back at my A level maths. In an algorithms module, we studied what is called eularian (I think thats how you spell it) maps. These are maps that you can map out in one line without going over an edge twice, for an ideal route. To cut through all the mathematical jargon, we worked out that to form a eularian map, axes had to have either 3 or 5 edges connected. JUST like 3D topological extrusion rigging. So the ideal mapping of a topological layout when deforming to create and ear or pit for an eye, is in fact a 3D eularian map.

INSPIRATION

Here is a link to Brian Silva's portfolio sight. It hasn't got any rigs in but the work is of a ridiculous standard. It is a great inspiration when working on my own models
www.bryansilva.com/index.html

STYLIZED

As you can see topology for the more stylized characters still follow the basic facial muscles layout. The one of Scrat from Ice Age however has a few minor differences because of its stylized shape. But it still follows the structure of a realistic squirrel, just manipulated.

EXAMPLE OF TYPICAL TOPOLOGICAL LAYOUT

As a comparison to my previous images uploaded, here is a a selection of the same topological layout but morphed to create emotion and expression. As you can see lines in the grid represent the basic structure of the muscles seen in my last post. A brilliant example of how helpful facial topology is when its come to 3D modeling and animation.

MORE IN DETAIL

Here is that same image again but with all the muscles labelled.

EXAMPLE OF MUSCLE STRUCTURE

Here is the basic muscular structure of the face. This is what all topological layouts are based on. Some may vary in structure, simply because of limitation or certain character traits which require specific type of layouts.

MATHS? I THOUGHT ID LEFT THAT IN COLLEGE!?

I found this short explanation on how facial topology mediates the decoding of facial expression.

It took me a couple of read throughs to completely understand it (still don't completely understand it, perhaps going slightly too mathematical)

Perception ECVP abstract

Intro

Topology is a major area of mathematics. Mapping out spacial properties that are preserved under continuous stretching, squashing and general deformation without breaking or tearing the topological layout.

Applying this to 3D modeling of a head, allows for an easier more natural manipulation of facial expression and emotion. This can also be very handy when following the 3D model up with animation.

I am going to use this blog to show you the development of topological 3D mapping through common debates about techniques and studying solid sources (Of course Im gonna try and not make it sound as boring as that) Enjoy :)