Three points define a circle exactly. That is why a short arc looks harmless on the drawing: the points are there, the software returns a diameter, the report shows a number. What the number does not show is how much of the tolerance the uncertainty has already taken.
Set the number of points, the arc of contact, the MPE of your machine and the tolerance. The graphic solves the covariance matrix of the least-squares circle directly, so the values are reproducible rather than simulated.
On a full circle the points constrain the form from every side. Centre and radius are determined independently of one another. On a short arc that separation collapses: a larger circle with a shifted centre describes the same points almost as well. Centre and radius become correlated, and the residual freedom lands in the diameter.
Adding points barely helps. The uncertainty falls with the square root of the number of points, while the geometry factor stays. Twenty-five points on 30° are still worse than nine points on 180°. What helps is opening the arc — or measuring the characteristic that the geometry actually carries.
Where a short arc is unavoidable, it belongs in the measurement process capability assessment before the first part is measured, not in the report afterwards.
A standard uncertainty of 63 µm means nothing on its own. It becomes a statement against the tolerance: the expanded uncertainty U = 2·u(D) compared with the tolerance span T. Below 30 percent the tolerance carries the measurement; that 30 percent mark is a convention common in measurement system analysis, not a requirement from the drawing. At 100 percent the uncertainty fills the span and no range remains in which conforming and non-conforming parts can be told apart safely.
| Arc · points | u(D) | U = 2·u(D) | 2U/T | Verdict |
|---|---|---|---|---|
| 360° · 9 | 0.80 µm | 1.60 µm | 8 % | inspectable |
| 180° · 9 | 6.62 µm | 13.2 µm | 66 % | marginally inspectable |
| 30° · 9 | 63.4 µm | 127 µm | 634 % | not inspectable |
| 30° · 25 | 43.2 µm | 86.5 µm | 432 % | not inspectable |
⌀ 20 mm with ±0.020 mm, MPEE 1.7 µm. At this tolerance the diameter needs roughly 75° of contact before the measurement fits inside it again. Cases like this show up in the drawing review, long before the part reaches the machine.
Our own measurement uncertainties are determined to ISO 15530 with calibrated workpieces. Alongside the theoretical value from this graphic that gives an experimentally determined value for the actual inspection process, carried out in our measuring laboratory.
The graphic derives the point uncertainty from the machine specification: up = MPEE/√2, formed from the maximum permissible error as is customary in measurement system analysis. The factor is a CSL convention for this consideration; where a project uses a different approach, that approach applies. Form deviation of the part is not included — it adds to the values shown.
