Abstract
This work addresses the problem of enclosing given data points between two concentric circles (spheres) of minimum distance whose associated annulus measures the out-of-roundness (OOR) tolerance. The problem arises in analyzing coordinate measuring machine (CMM) data taken against circular (spherical) features of manufactured parts. It also can be interpreted as the “geometric” Chebychev problem of fitting a circle (sphere) to data so as to minimize the maximum distance deviation. A related formulation, the “algebraic” Chebychev, determines the equation of a circle (sphere) so as to minimize the maximum violation of this equation by the data points. The algebraic Chebychev amounts to a straightforward linear-programming problem. In this paper, we compare the algebraic Chebychev against the popular algebraic least-squares solutions for various data sets. In most of these examples, the algebraic and geometric Chebychev solutions coincide, which appears to be the case for most real applications. Such solutions yield concentric circles whose separation is less than that of the corresponding least-squares solution. It is suggested that the linear-programming approach be considered as an alternate solution method for determining OOR annuluses for CMM data sets.
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References
American Society of Mechanical Engineers, 1972. Measurement of Out-Of-Roundness, ANSI B89.3. 1–1972, NY, NY.
Anthony, G. T., H. M. Anthony, B. Bittner, H. P. Butler, M. G. Cox, R. Drieschner, R. Elligsen, A. B. Forbes, H. Gross, S. A. Hannaby, P. M. Harris and J. Kok, 1993. Chebychev Best-Fit Geometric Elements, NPL Report DITC 22193. National Physical Laboratory, Teddington, United Kingdom.
Barber, C. B., D. P. Dobkin and H. T. Huhdanpaa, 1995, The Quickhull Algorithm for Convex Hulls, University of Minnesota.
Boggs, P. T., R. H. Byrd, J. E. Rogers and R. B. Schnabel, 1992. User’s Reference Guide for ODRPACK Version 2.1 - Software for Weighted Orthogonal Distance Regression, National Institute of Standards and Technology, Gaithersburg, Maryland.
Butler, B. P., A. B. Forbes and P. M. Harris, 1994. Algorithms for Geometric Tolerance Assessment, NPL Report DITC 228/94, National Physics Laboratory, Teddington, United Kingdom.
Carr, K., and P. Ferreira, 1994. Verification of Form Tolerances, Part I: Basic Issues, Flatness and Straightness, and Part II: Cylindricity, Cirularity, and Straightness of a Median Line, Ford Technical Report SR-94–14, Ford Research Laboratory, Flint, Michigan.
Chetwynd, D. G., 1985. Applications of Linear Programming to Engineering Metrology, Proceedings, Institute of Mechanical Engineers, Vol. 199, No. B2, 93–100.
Chou, S.-Y., T. C. Woo and S. M. Pollock, 1994. On Characterizing Circularity, Department of Industrial and Operations Research, University of Michigan, Ann Arbor.
Elzinga, D J, and D. W. Hearn, 1972. The Minimum Covering Sphere Problem, Management Science, 19, 1, 96–104.
Etesami, F., and H. Qiao, 1988. Analysis of Two-Dimensional Measurement Data for Automated Inspection, Journal of Manufacturing Systems, 7, 3, 223232.
Feng, S. C., and T. H. Hopp, 1991. A Review of Current Geometric Tolerancing Theories and Inspection Data Analysis Algorithms, NISTIR 4509, National Institute of Standards and Technology, Gaithersburg, Maryland.
Gander, W., G. H. Golub, and R. Strebel, 1994. Least-Squares Fitting of Circle and Ellipses, BIT, 24, 560–578.
Gass, S. I., 1985. Linear Programming, McGraw-Hill Book Company, NY, NY.
Harary, H., J.-O. Dufraigne, P. Chollet and A. Clement, 1993. Probe Metrology and Probing with Coordinate Measuring Machines, International Journal of Flexible Automation and Integrated Management, 1, 1, 59–70.
Hearn, D. W., and J. Vijay, 1982. Efficient Algorithms for the (Weighted) Minimum Circle Problem, Operations Research, 30, 4, 777–795.
Hopp, T. H. and C. A. Reeve, 1996. An Algorithm for Computing the Minimum Covering Sphere in any Dimension, NISTIR 5831, National Institute of Standards and Technology, Gaithersburg, MD.
Jones, B. A., and R. B. Schnabel, 1986. A Comparison of Two Sphere Fitting Methods, Proceedings of the Instrumentation and Measurement Technology, Subgroup of the IEEE, Boulder, Colorado.
Le, V.-B., and D. T. Lee,1991. Out-of-Roundness Problem Revisited, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 3, 217–223.
Lee, D. T., and R. L. Drysdale, III, 1081. Generalization of Vornoi Diagrams in the Plane, Siam Journal of Computing, 10, 1, 73–87.
Megiddo, N., 1983 Linear-Time Algorithms for Linear Programming in R3 and Related Problems, Siam Journal of Computing, 12, 4, 759–776.
Preparata, F. P. and M. I. Shamos, 1985. Computational Geometry, Springer-Verlag, New York.
Phillips, S. D., 1995. Performance Evaluation, Chapter 7 in Coordinate Measuring Machines and Systems, edited by J. Bosch, Marcel Dekker, 137–220.
Phillips, S. D., B. Borchardt, and G. Caskey, 1993. Measurement Uncertainty Considerations for Coordinate Measuring Machines, NISTIR 5170, National Institute of Standards and Technology, Gaithersburg, Maryland.
Shannos, M. I, and D. Hoey, 1975. Closest-Point Problems, Proceedings of the 16th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Long Beach, California, 151–162.
Schrage, L.,1989.User’s Manual: Linear Integer and Quadratic Programming with Lindo, 4th Edition, Scientific Press.
Zhou, J. L., A. L. Tits and C. T. Lawrence, 1995. FFSQ Software, Electrical Engineering Department, University of Maryland, College Park, Maryland.
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Gass, S.I., Witzgall, C., Harary, H.H. (1998). Fitting Circles and Spheres to Coordinate Measuring Machine Data. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds) New Trends in Mathematical Programming. Applied Optimization, vol 13. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2878-1_7
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DOI: https://doi.org/10.1007/978-1-4757-2878-1_7
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