I am working on writing a patent for which I am not free to disclose the details, so will only disclose a limited amount that I hope will still be sufficient to compose a useful question.

The invention requires the careful orientation of a pivoting axis so that it doesn't collide with another element. In prior art this has been done with a pivot that is simply square with the structure, but there are some advantages to altering the pivot orientation to being skew to the structure. In the claims, I think I've defined this clearly enough, by defining it as being skew to the structure while still permitting a sufficient angle of rotation. My question, however, relates to the specification, where I describe in more detail the particular pivot orientations that will work to enable sufficient rotation angle.

My understanding is that the specification must include detailed instructions explaining how to build any embodiment of the invention. I have several drawings of different embodiments, demonstrating various pivot orientations that will work to rotate far enough without collision. However, I am wondering into how much mathematical detail I need to go to describe instructively which pivot orientations will enable sufficient rotation. When creating the embodiments I have drawn, I first just created them in CAD using more of a trial-and-error approach, but also, later, used some fairly sophisticated mathematics (quaternions, and advanced geometry), to compute exact pivot orientations that gave clean drawings. I am wondering whether this math should be included as instruction in the specification.

Currently in the specification, to describe how a working orientation angle can be determined, I say something like, "to identify suitable pivot orientations, one can construct the components in CAD, parameterizing the location and orientation of the pivots, and modify the parameters to determine which will suffice to obtain the desired rotation without collision." While not the only way to identify appropriate pivots, I do think that any potential embodiment could be made using that method. Is this sufficient of an explanation, per the standards of a U.S. patent?

Alternatively, I could include a description of some of the mathematics that I've used to determine some pivot axes more exactly, but going into any kind of detail on that would be fairly technical. Also I'm not sure it would be adding anything meaningful, since perhaps the drawings and basic description as defined in the claims and CAD suggestion is instructive enough on how to build. I could just kind of gloss over the math in broad terms, if that would be best.

Do I need to go into detail describing the math that may help to identify exact pivot orientations? On one hand, I must clearly give instructions sufficient to build any embodiment. On the other hand, since the patent is intended to cover the ultimate invention, it should not need instructions on more than one possible method of designing it. Is this on the right track in how to even think about this?