Uncertainty of extreme fit evaluation for three-dimensional measurement data analysis Woncheol Choi and Thomas R Kurfess* Three-dimensional measurement is the process of obtaining information about a measured object in the form of surface coordinates. Of course this information must be accompanied by a data analysis procedure that evaluates geometric dimensions from the measured data. Currently extreme fits, that are based on an L  norm estimation, are widely applied to analyze these data. The extreme fit results reflect the functionality of the part, and they are more consistent with standard definitions. However, they are subject to sampling uncertainty as they are sensitive to measure- ment sampling density. Previous studies focused on the evaluation methodology of the extreme fits; however, the uncertainty of the extreme fits has not been thoroughly addressed. In this paper, we investigate the uncertainty of extreme fits, and propose a statistical approach to evaluate it. We employ the bootstrap method to determine the confidence interval of the extreme fit evaluations. The proposed method is independent of the geometry and evaluation method; thus, it can be easily generalized for various extreme fit evaluations and geometries.  1998 Elsevier Science Ltd. All rights reserved Keywords: metrology, tolerance, precision, manufacturing, quality, sampling INTRODUCTION With the implementation of three-dimensional measure- ment capabilities, metrology systems have become sig- nificantly more powerful. A three-dimensional measurement device such as a coordinate measuring machine (CMM) samples the surface of a measure object and generates three-dimensional coordinate points from that surface. The measurement output is a collection of digitized surface points. Compared to traditional two-point measure- ments, the three-dimensional measurement yields more comprehensive information about the actual part geometry. Thus, more details about surface variation can be observed, and various sections of different geometry features can be measured in a single process. Currently, three-dimensional measurement devices are an essential tool for measuring a free form surface (e.g. turbine blades), and they can serve as a general purpose dimensional measurement device as well. After measurement data are collected by a three- dimensional measuring machine, an independent numerical analysis must be performed to evaluate geometric variations or verify tolerance conformance. The measurement output for all such measurement systems is processed in a compu- tational environment as a geometric rather than a parametric entity. Being in a geometric form has the advantage that the measurement output can be directly integrated with CAD design models. On the other side, one must process the measured data to extract parametric information. The measured data are typically a set of vectors representing the coordinates of surface points. If one is interested in evaluating the individual deviations of the measured points, an appropriate rigid body transformation must be applied to the point set, and then the deviation from the design model must be obtained. This task is known as data localization. On the other hand, if one is interested in a specific geometric value, such as the diameter of a hole feature or a B-spline surface representing the surface variation, then an appro- priate geometry must be reconstructed from the measured points. As such, the focus of the numerical analysis is to transform the raw coordinate data into a more appropriate form for interpretation (i.e. inspection results). Least squares and extreme fits are typically utilized in coordinate data numerical analysis and various researchers have developed a number of numerical algorithms for three- dimensional data analysis. Most CMMs include data analy- sis software using the least squares fit. Examples of least squares and extreme fits are shown in Figures 1 and 2. Recently, the extreme fit evaluation has generated signifi- cant interest by metrologists as it represents an L  norm estimation such as min–max fit. The main advantage for the extreme fit is that it is more consistent with ASME dimensioning and tolerancing standards 1,2 . The geometric concepts defined in the standards, such as actual mating envelope or circularity, can only be verified by utilizing extreme fits as shown in Figure 2. The use of the extreme fit poses a new set of challenging issues related to the inaccuracy of the data analysis results. As recent studies demonstrate 3–5 , current data analysis tech- niques are not robust and implicitly possess uncertainty. Furthermore, the three-dimensional measurement data analysis procedures are not a closed-form evaluation proce- dure; rather, they are a numerical iterative estimation that Computer-Aided Design, Vol. 30, No. 7, pp. 549–557, 1998  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0010-4485/98/$19.00+0.00 PII: S0010-4485(98)00012-8 549 *To whom all correspondence should be addressed The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Paper Received: 9 December 1996. Revised: 1 February 1998. Accepted: 1 February 1998