Calibration Using a Manifold and ?• Manifold

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Abstract

A manifold is a measurement space. In small regions it is straight, square, and linear, which is commonly called Euclidean, but a manifold will often be more complex over larger regions, that is, not straight, square, and linear across large segments of the measurement volume. ?• Manifold is the mathematically shorthand for “divergence on a manifold.” The divergence quantifies the changes in the measurement space in independent directions, so in small regions of space the value of ?• Manifold would be uniform, but across larger regions will be slightly different in different directions. Calibration of an instrument entails comparison of measurements to a standard and the estimation of the uncertainty for those measurements. Generally the calibration for 3D measurement systems takes one of two forms: measurement of locations within a volume, and measurements of known lengths within a volume. Measurements within a volume can either be known-invariant values with a volume measured from one position, or unknown-invariant values with some method of determining scale measured from several positions. Measurements of known lengths tend to be measurements from a single instrument position and rely on multiple positions and orientations of the length within that volume. This article examines these two methods.