Measurerement vs. theoretical element: Theory of metrology and my confusion

I have recently joined a small quality department in a starting company. We try to do as many measurements as possible on CMM. Last week, we measured a diameter on a series of pieces, many of which turned out to be nonconforming. There was no circularity check in the part drawing, so we only found out later when analyzing that the pieces were clamped too hard during machining and got elliptical (aluminum alloy).

It was Friday (brains fried) and we found ourselves baffled at this realization:
When you measure a diameter with a caliper, you repeat the measurement at different positions in relation to the circumference ("at angles") and average it. This is a direct measurement (is it?) which can also reveal an issue with the circularity. But then, through a train of thought, imagine that that circularity is completely out of bounds — the shape of the hole is a mess, and you still perform the measurements, and average them. Question #1: How does the measured "diameter" relate to the theoretical element of circle? How to interpret the measured diameter? You could be measuring a square hole and the circumference could come out fine. I can't get my head over it. This issue stands even if you would describe the hole that's to be measured as "circular enough". Diameter is a property of a circle, and you're measuring a real linear dimension that doesn't relate to any REAL circle.

Question #2: How does this translate to CMM? You measure some points on the """circumference""" and use something like the smallest squares method to find the approximate circle. Again, how does the diameter of this theoretical circle relate to the real thing?

We do have a "general" tolerances that apply to unquoted dimensions and an acceptable circularity is actually defined, but that's not the issue here.


I'm sorry if I'm being too dense. We realized that we may be overthinking the whole thing, but then again, we would like to get this straight, more so because we don't have too much training in theoretical metrology. We were thinking — when we measure a diameter, should we also evaluate the related circularity? Is there any literature you guys would recommend to get me out of this Lovecraftian madness? Thank you for your time...

 

Have you ever seen a print that identified how a diameter should be evaluated (ie: gaussian least squares, maximum inscribed, minimum circumscribed, inner tangential, outer tangential, etc)? Forty year in quality and I haven't. Furthermore, when the customer is queried about what's appropriate for their application, most have been unable to provide any guidance.

 

I don't know what software package you have but this Zeiss presentation should help you understand how your algorithm selection affects the measurement.