Hi Richard. Sorry, I must be dense. I’m including images to fill in maybe some of the 1000 words. 
Aside from what is obviously Horse 2, is my halter and browband which are separate items. these have been fit to Horse 2.
I’m not sure I understand the difference between a morph and a transform? The Daz docs on Rigidity maps say:
““Rigidity” is a function that protects areas of mesh from distortion when morphs are automatically generated to match a TriAx™ figure to which the item has been configured. Think of Rigidity as “Resistance to morphing.” An example would be a metal ring joining the two sides of a bikini top. If the user applies a morph to change the size of the figure’s breasts, Rigidity would allow the metal ring to remain round even though the dynamically projected morph might be otherwise. Rigidity is also useful for buttons, pockets, buckles, and anything else that should maintain its shape or position relative to a reference point on the rest of the mesh, regardless of morphs applied to the underlying character. “
Is a breast PBM morph different from a Horse 2 head FBM? Well, aside from the obvious.
Both are shown in the params as Type: Modifier/Shape. I have assumed BM stands for body morph? I’m not sure what P or F stand for.
Anyway, the halter below, like a bikini top mentioned above, has various hardware or ‘a ring’ to hold 2 (or more) sides together. I’m trying to stop the distortion of these metal bits. I thought I understood that the underlying triax would move the vertices around even if a full weight map was applied?
The non-distorted image is Horse 2. The image with distortion has the Arabian Breed set to full 1.00.
I have a rigidity map set on the medallion (coin shape) which is a part of the browband. Otherwise, this distorts badly. I also have a rigidity map set on the ring under the chin and the 3-way ring on the nose. The buckles obviously need work. I sure hope I don’t have to fuss with all of these polys for each breed to fix this via the morphs. Whew!!!! As per your help in another thread, I have completed my reins not pictured, by correcting via the Hex bridge and morph loader pro. But they need to remain flexible, just not wrinkled or distorted.
As for rotation. The same Daz Rigidity doc talks a bit about rotation, but doesn’t really go much beyond only a warning. I see in the Rigidity Groups Editor, a line for assigning Rotation, but none of the selections there hold after I save the item, but instead reverts back to No Rotation every time.
The Llama for instance needs to have rotation allowed badly. On the Arabian, you can’t see it in the image below, but the 3-way ring is not tilting properly and needs to rotate a bit in 2 axes, as it is cutting into the back of the halter bands.
I hope this helps better explain my situation. I do hope what I think I’m reading about rigidity maps is applicable in this situation, otherwise there is likely days of fixing the polys on these breed morphs.
And Richard… yet again thanks for being you. You are the best and have helped me a ton.

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