Uncertainty analysis for angle calibrations using circle closure

We analyze two types of full-circle angle calibrations: a simple closure in which a single set of unknown angular segments is sequentially compared with an unknown reference angle, and a dual closure in which two divided circles are simultaneously calibrated by intercomparison. In each case, the constraint of circle closure provides auxiliary information that (1) enables a complete calibration process without reference to separately calibrated reference artifacts, and (2) serves to reduce measurement uncertainty. We derive closed-form expressions for the combined standard uncertainties of angle calibrations, following guidelines published by the International Organization for Standardization (ISO) and NIST. The analysis includes methods for the quantitative evaluation of the standard uncertainty of small angle measurement using electronic autocollimators, including the effects of calibration uncertainty and air turbulence.