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@j.palmeL29YX , Yes, the video works great—thank you! The errors in the geometry that you point out come from rotating around the y-axis of either the WCS or UCS (I kept switching between them regularly), where the y-axes of the two coordinate systems made a very small angle, on the order of 0.00004 radians. So, when I rotated by -90 around one of those y-axis, then in the other coordinate system I had a 269.9999xxxx angle, which rounded to 270 (escaping my error checking), but causing all of the issues I had here.
It is true that if I had used the 3-point 3dalign procedure as you did, everything would have worked. However, then my object would be at those weird angles, and I am not sure that my complete system (much more than shown in that drawing) would work with me switching back and forth all of the time between the WCS and my UCS. As in, maybe I could close the gaps, but eventually I was going to make a closed gasket, and I am not sure the final ends would touch to create the closed loop, or that that loop would sit right on my other parts, which might have been aligned with the opposite coordinate system. Not sure.
Anyway, for me the big issue was relying on the 4 decimals displayed when there are really 8 (please tell me there are not 9-16 that cannot be shown or something, which also matter for things like Union etc.).