This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

Issue Section:
Research Papers
Keywords:
Computational geometry, CMM, GD&, Tolerance modeling
Topics:
Cylinders, Fittings

References

1.
ASME,
2009
, “
Dimensioning and Tolerancing
,” American Society of Mechanical Engineers, New York, Standard No.
Y14.5-2009
.https://www.asme.org/products/codes-standards/y145-2009-dimensioning-and-tolerancing
2.
ASME,
1994
, “
Mathematical Definition of Dimensioning and Tolerancing Principles
,” American Society of Mechanical Engineers, New York, Standard No.
Y14.5.1-1994
.https://www.asme.org/products/codes-standards/y1451m-1994-mathematical-definition-dimensioning
3.
ISO
,
2012
, “
Geometrical Product Specifications (GPS)—Geometrical Tolerancing—Tolerances of Form, Orientation, Location and Run-Out
,” International Organization for Standardization, Geneva, Switzerland, Standard No.
ISO 1101:2012
.https://www.iso.org/standard/59490.html
4.
ISO
,
2011
, “
Geometrical Product Specifications (GPS)—Geometrical Tolerancing—Datums and Datum Systems
,” International Organization for Standardization, Geneva, Switzerland, Standard No.
ISO 5459:2011
.https://www.iso.org/standard/40358.html
5.
Nielsen
,
H. S.
,
2012
, “
The ISO Geometrical Product Specifications Handbook—Find Your Way in GPS
,” International Organization for Standardization, Geneva, Switzerland.
6.
Krulikowski
,
A.
,
2012
,
Fundamentals of Geometric Dimensioning and Tolerancing
, 3rd ed.,
Effective Training
,
Westland, MI
.
7.
Srinivasan
,
V.
,
2014
, “
Geometrical Product Specification
,”
CIRP Encyclopedia of Production Engineering
,
L.
Laperrière
 and
G.
Reinhart
, eds.,
Springer
, Berlin.
8.
Wick
,
C.
, and
Veilleux
,
R. F.
,
1987
,
Tool and Manufacturing Engineers Handbook: Quality Control and Assembly
, 4th ed., Vol.
4
,
Society of Manufacturing Engineers
,
Dearborn, MI
.
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.
,
2016
, “
Computational Investigations for a New, Constrained Least-Squares Datum Definition for Circles, Cylinders, and Spheres
,”
ASME
Paper No. IMECE2016-66753.
12.
Gill
,
P. E.
,
Murray
,
W.
, and
Wright
,
M. H.
,
1982
,
Practical Optimization
,
Emerald Group Publishing Ltd
., Bingley, UK.
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.
Hopp
,
T. H.
,
1990
, “
The Mathematics of Datums
,” ASPE Newsletter, The American Society of Precision Engineering, Raleigh, NC.
15.
Zhang
,
H.
, and
Rou
,
U.
,
1993
, “
Criteria for Establishing Datums in Manufactured Parts
,”
J. Manuf. Syst.
,
12
(
1
), pp.
36
–
50
.
Google Scholar
Crossref
Search ADS
 
16.
Anthony
,
G. T.
,
Anthony
,
H. M.
,
Bittner
,
B.
,
Butler
,
B. P.
,
Cox
,
M. G.
,
Drieschner
,
R.
,
Elligsen
,
R.
,
Forbes
,
A. B.
,
Gross
,
H.
,
Hannaby
,
S. A.
,
Harris
,
P. M.
, and
Kok
,
J.
,
1993
, “
Chebyshev Best-Fit Geometric Elements
,” National Physical Laboratory, Teddington, UK, NPL Report No. DITC 221/93.
17.
Butler
,
B. P.
,
Forbes
,
A. B.
, and
Harris
,
P. M.
,
1994
, “
Algorithms for Geometric Tolerance Assessment
,” National Physical Laboratory, Teddington, UK, NPL Report No. DITC 228/94.
18.
Srinivasan
,
V.
,
2013
, “
Reflections on the Role of Science in the Evolution of Dimensioning and Tolerancing Standards
,”
Proc. Inst. Mech. Eng., Part B
,
227
(
1
), pp.
3
–
11
.
Google Scholar
Crossref
Search ADS
 
19.
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.
20.
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.
21.
Shakarji
,
C. M.
, and
Srinivasan
,
V.
,
2013
, “
Theory and Algorithms for L1 Fitting Used for Planar Datum Establishment in Support of Tolerancing Standards
,”
ASME
Paper No. DETC2013-12372.
22.
Shakarji
,
C. M.
, and
Srinivasan
,
V.
,
2014
, “
An Improved L1 Based Algorithm for Standardized Planar Datum Establishment
,”
ASME
Paper No. DETC2014-35461.
23.
Shakarji
,
C. M.
, and
Srinivasan
,
V.
,
2015
, “
Datum Planes Based on a Constrained L1 Norm
,”
ASME J. Comput. Inf. Sci. Eng.
,
15
(
4
), p. 041008.
24.
O'Rourke
,
J.
,
1998
,
Computational Geometry in C
, 2nd ed.,
Cambridge University Press
, Cambridge, UK.
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