This report describes algorithms for fitting certain curves and surfaces to points in three dimensions. All fits are based on orthogonal distance regression. The algorithms were developed as reference software for the National Institute of Standards and Technology’s Algorithm Testing System, which has been used for 5 years by NIST and by members of the American Society of Mechanical Engineers’ B89.4.10 standards committee. The Algorithm Testing System itself is described only briefly; the main part of this paper covers the general linear algebra, numerical analysis, and optimization methods it employs. Most of the fitting routines rely on the Levenberg-Marquardt optimization routine.

Two hundred eighty three uniaxial ellipsoids with sizes from 4 mm to 11 mm were measured with a coordinate measuring matching (CMM) and also scanned using a medical computed tomography (CT) machine. Their volumes were determined by counting voxels over a threshold, as well as using equivalent volumes from the length given by the RECIST 1.1 criterion (Response Evaluation Criteria in Solid Tumors). The volumetric measurements yield an order of magnitude reduction in residuals compared to the CMM measurements than the residuals of the RECIST measurements also compared to the CMM measurements.

We present the theory and algorithms for fitting a line, a plane, two parallel planes (corresponding to a slot or a slab), or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3 × 3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact coordinate measuring machine). However, we show by example that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

This paper describes a threefold method of testing the performance of an array-based ultrasonic tool for nondestructive testing of spot welds. The tool is described in its capabilities, use, and advantages over existing counterparts. Performance testing for and the results from carrying out the testing are described. The three performance testing methods include 1) the use of calibrated samples, 2) comparisons with actual spot-welds, and 3) a performance evaluation of the embedded fitting software. The test of the fitting software was carried out by a comparison of results with reference fits supplied by the National Institute of Standards and Technology.