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Research Papers

On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

[+] Author and Article Information
Craig M. Shakarji

Mem. ASME
Physical Measurement Laboratory,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: craig.shakarji@nist.gov

Vijay Srinivasan

Fellow ASME
Engineering Laboratory,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: vijay.srinivasan@nist.gov

Manuscript received September 14, 2018; final manuscript received March 5, 2019; published online May 15, 2019. Assoc. Editor: Yan Wang. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Comput. Inf. Sci. Eng 19(3), 031012 (May 15, 2019) (12 pages) Paper No: JCISE-18-1240; doi: 10.1115/1.4043226 History: Received September 14, 2018; Revised March 05, 2019

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two dimensions) and spheres (in three dimensions) using constrained least-squares criterion to a set of points that are usually described as a “point-cloud.” This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

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Copyright © 2019 by ASME
Topics: Algorithms , Fittings
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References

Shakarji, C. M. , and Srinivasan, V. , 2018, “ Toward a New Mathematical Definition of Datums in Standards to Support Advanced Manufacturing,” ASME Paper No. MSEC2018-6305.
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Srinivasan, V. , Shakarji, C. M. , and Morse, E. P. , 2012, “ On the Enduring Appeal of Least-Squares Fitting in Computational Coordinate Metrology,” ASME J. Comput. Inf. Sci. Eng., 12(1), p. 011008. [CrossRef]
Shakarji, C. M. , and Srinivasan, V. , 2013, “ Theory and Algorithms for Weighted Total Least-Squares Fitting of Lines, Planes, and Parallel Planes to Support Tolerancing Standards,” ASME J. Comput. Inf. Sci. Eng., 13(3), p. 031008. [CrossRef]
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Golub, G. H. , and Matt, U. V. , 1991, “ Quadratically Constrained Least Squares and Quadratic Problems,” Numerische Math., 59(1), pp. 561–580. [CrossRef]
Peng, J. J. , and Liao, A. P. , 2017, “ Algorithm for Inequality-Constrained Least Squares Problems,” Comput. Appl. Math., 36(1), pp. 249–258. [CrossRef]
Shakarji, C. M. , and Srinivasan, V. , 2015, “ A Constrained L2 Based Algorithm for Standardized Planar Datum Establishment,” ASME Paper No. IMECE2015-50654.
Shakarji, C. M. , and Srinivasan, V. , 2016, “ Theory and Algorithm for Planar Datum Establishment Using Constrained Total Least-Squares,” 14th CIRP Conference on Computer Aided Tolerancing, Gothenburg, Sweden, May 18–20, pp. 232–237.
Shakarji, C. M. , and Srinivasan, V. , 2017, “ Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums,” ASME Paper No. IMECE2017-70899.
Shakarji, C. M. , and Srinivasan, V. , 2018, “ Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums,” ASME J. Comput. Inf. Sci. Eng., 18(3), p. 031008. [CrossRef]
Shakarji, C. M. , and Srinivasan, V. , 2017, “ Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums,” ASME Paper No. DETC2017-67143.
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Shakarji, C. M. , and Srinivasan, V. , 2018, “ On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards,” ASME Paper No. DETC2018-85109.

Figures

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Fig. 1

Illustration of notations for the optimization problem

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Fig. 2

Objective function for Example E1

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Fig. 3

Contour plot of the objective function of Fig. 2

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Fig. 4

Voronoi diagram of input points in Example E1

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Fig. 5

Superposition of Figs. 3 and 4

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Fig. 6

Objective function along the five Voronoi edges shown in Figs. 4 and 5. The bottom-most function is along the only short, finite Voronoi edge. The other four functions are along the half-line Voronoi edges that have been trimmed.

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Fig. 7

Results from A_CL2IC for Example 1

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Fig. 8

Results from A_CL2IC for Example 2

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Fig. 9

Parabola and parallel lines

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Fig. 10

Results from H_CL2IC for Example 1

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Fig. 11

Results from H_CL2IC for Example 2

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Fig. 12

Results from H_CL2CC for Example 1

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Fig. 13

Results from H_CL2CC for Example 2

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