You would not want to rig this little beast.

A character made from one piece of geometry can not be rigged if:-

it is out of alignment with the z axis
it is non manifold.
some faces have inverse surface normals.
the UV coordinates are untidy or share the same UV space.
the translate & rotate transformations are not zero or if the scale transformations are not set to 1.0
there is any constructional or tweek history.
nodes are not properly named. (“polySurface203” is not appropriate).(“ear_left” is fine).

A character made from more than one piece of geometry can not be rigged if:-

each piece of character does not abide by the rules above.
elements of geometry are not placed in a well ordered and logical hierarchy.
geometry transform nodes are nested within each other. (They should be placed within group using null transform nodes.)
the hierarchical structure of the Blend Shape targets are not all exactly the same and do not match the original hierarchy.

If the rigging is implemented without taking account of the above then, in some cases, it might still work but your scene file may become unstable especially while rendering.   In some circumstances it can also slow your render down. In the worst case it can break your scene so that it becomes inoperable.

If you want to know why all these measures are necessary then read the “NOTES” below.

Trouble Shooting Geometry.

Test you knowledge by downloading and examining two scene files.   Imagine you are a TD and you are responsible for character rigging but before you make a start your must check that these characters are faultless.

What's Wrong with these Six Dogs
Free to Download

Seven Stag Heads:
In the first scene file there is the geometry of 7 stag heads.   Apart from being very poorly designed and not very well built there are technical problems with the geometry of one or more of them.   See if you can find out what these faults are.   Are any of them are faultless and fit to be rigged?   If you are not sure what to do then first try smoothing the geometry by adding a smooth node, this can often amplify vaults.   Other tools that can help are the polygon display tools like showing surface normals and border edges.

What's Wrong with this Dog           Free to Download

One Stag Head:
In the second scene file there is the geometry of one stag head.   There are no problems with its geometry but are there any reasons why this head is unfit to rig?
I can think of at least five.
To discourage you from cheating, the answers to these questions can only be viewed in a .pdf which can be downloaded for free from the download section of this site.

NOTES: Why are all these measures necessary?
Rigging can be difficult if the geometry is asymmetrical.

None manifold geometry at worst, can cause division by zero at render time and that can cause a render to stop and at best cause holes or creases in your surfaces.

Surface normals that point the wrong way can cause unwanted artifacts when you attempt to light them.

You can’t apply 2d textures to geometry that has untidy or overlapping UVs.

Why do translate & rotate transformations have to be zeroed and scales set to 1.0?   If a transformation is not part of your animation they should be frozen which removes translations, rotations and scalings and resolves the geometry at it’s absolute position.   Each transformation node takes time to compute at render time but if a transform has been reset, it sets a flag.   During the render, it looks at the flag and if it is set then it doesn’t bother to visit the transform node, thus saving a calculation.   The amount of time that is saved in this way is not much but it does cut down on mathematic error and therefore adds to stability of your scene file.

Why delete history?.
Everything you produce in Maya produces at least one node.   A node is a thing that does something and it can be connected to other nodes via inputs and outputs into long strings or networks called a DG or Dependency Graph. When you create a Polygon Cube you create two nodes.   One is a transformation node used for implementing translation, rotations and scaling etc and other is a shape node which describes the geometry.   When you change this geometry by extruding a new face or inserting a new edge or any of the other ways of modifying, it doesn’t actually do it straight away, but merely adds a new node that says it’s going to do it.   Weird I know but if you don’t delete the history of the geometry construction then you end up with endless chains of these construction nodes.   When you render, the DG has to be resolved for each frame and this means the renderer has to check all the way “upstream” through all the nodes before it can commit to what the shape of the geometry actually is.  If you delete history from geometry then the constructional nodes are removed and the geometry resolved.   The renderer can go straight to the shape node via the transform node which is more efficient.   Why have history in the first place?   Actually, that is what makes Maya extremely powerful and flexible.   There are certain conditions when leaving constructional history in place can be beneficial.

Pivot points should be placed in a meaningful positions even if their are not essential for the animation. They should be set to either the origin or the object centre.   A pivot point which stands away from the geometry can be confusing when the geometry is selected.

Nodes not properly named causes conflict or convolution when using expressions and will cause havoc when scene files are shared between team members.

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