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.

References

References
1.
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.
2.
Forbes
,
A. B.
,
1991
, “
Least-Squares Best-Fit Geometric Elements
,” National Physical Laboratory, Teddington, UK, Report No.
DITC 140/89
.https://www.researchgate.net/publication/274371520_Least-squares_best-fit_geometric_elements
3.
Shakarji
,
C. M.
,
1998
, “Least-Squares Fitting Algorithms of the NIST Algorithm Testing System,”
J. Res. Natl. Inst. Stand. Technol.
,
103
(
6
), pp.
633
641
.https://nvlpubs.nist.gov/nistpubs/jres/103/6/j36sha.pdf
4.
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
.
5.
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
.
6.
Coons
,
S. A.
,
1978
, “
Constrained Least-Squares
,”
Comput. Graph.
,
3
(
1
), pp.
43
47
.
7.
Golub
,
G. H.
, and
Matt
,
U. V.
,
1991
, “
Quadratically Constrained Least Squares and Quadratic Problems
,”
Numerische Math.
,
59
(
1
), pp.
561
580
.
8.
Peng
,
J. J.
, and
Liao
,
A. P.
,
2017
, “
Algorithm for Inequality-Constrained Least Squares Problems
,”
Comput. Appl. Math.
,
36
(
1
), pp.
249
258
.
9.
Shakarji
,
C. M.
, and
Srinivasan
,
V.
,
2015
, “
A Constrained L2 Based Algorithm for Standardized Planar Datum Establishment
,”
ASME
Paper No. IMECE2015-50654.
10.
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.
11.
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.
12.
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
.
13.
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.
14.
O'Rourke
,
J.
,
1998
,
Computational Geometry in C
,
2nd ed.
,
Cambridge University Press
,
Cambridge, UK
.
15.
Barber
,
C. B.
,
Dobkin
,
D. P.
,
Dobkin
,
D. P.
, and
Huhdanpaa
,
H.
,
1996
, “
The Quickhull Algorithm for Convex Hulls
,” ACM Trans. Math. Software (TOMS),
22
(4), pp. 469–483.
16.
Eaton
,
J. W.
,
Bateman
,
D.
,
Hauberg
,
S.
, and
Wehbring
,
R.
,
2017
, “
GNU Octave 4.2 Reference Manual
,” Samurai Media Ltd., Thames Ditton, UK.
17.
Gill
,
P. E.
,
Murray
,
W.
, and
Wright
,
M. H.
,
1982
,
Practical Optimization
,
Emerald Group Publishing
,
Bingley, UK
.
18.
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.
You do not currently have access to this content.