Digital measurement devices, such as coordinate measuring machines, laser scanning devices, and digital imaging, can provide highly accurate and precise coordinate data representing the sampled surface. However, this discrete measurement process can only account for measured data points, not the entire continuous form, and is heavily influenced by the algorithm that interprets the measured data. The definition of cylindrical size for an external feature as specified by ASME Y14.5.1M-1994 [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] matches the analytical definition of a minimum circumscribing cylinder (MCC) when rule no. 1 [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] is applied to ensure a linear axis. Even though the MCC is a logical choice for size determination, it is highly sensitive to the sampling method and any uncertainties encountered in that process. Determining the least-sum-of-squares solution is an alternative method commonly utilized in size determination. However, the least-squares formulation seeks an optimal solution not based on the cylindrical size definition [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] and thus has been shown to be biased [Hopp, 1993, “Computational Metrology,” Manuf. Rev., 6(4), pp. 295–304; Nassef, and ElMaraghy, 1999, “Determination of Best Objective Function for Evaluating Geometric Deviations,” Int. J. Adv. Manuf. Technol., 15, pp. 90–95]. This work builds upon previous research in which the hull normal method was presented to determine the size of cylindrical bosses when rule no. 1 is applied [Turek, and Anand, 2007, “A Hull Normal Approach for Determining the Size of Cylindrical Features,” ASME, Atlanta, GA]. A thorough analysis of the hull normal method’s performance in various circumstances is presented here to validate it as a superior alternative to the least-squares and MCC solutions for size evaluation. The goal of the hull normal method is to recreate the sampled surface using computational geometry methods and to determine the cylinder’s axis and radius based upon it. Based on repetitive analyses of random samples of data from several measured parts and generated forms, it was concluded that the hull normal method outperformed all traditional solution methods. The hull normal method proved to be robust by having a lower bias and distributions that were skewed toward the true value of the radius, regardless of the amount of form error.

1.
Voelcker
,
H. B.
, 2002, “
Whither ‘Size’ in Geometric Tolerancing
,”
Proceedings of the ASPE Summer Topical Meeting on Tolerancing Modeling and Analysis
, pp.
85
90
.
2.
The American Society of Mechanical Engineers
, 1995,
Dimensioning and Tolerancing, ASME Standard Y14.5M-1994
,
ASME
,
New York, NY
.
3.
The American Society of Mechanical Engineers
, 1995,
Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994
,
ASME
,
New York, NY
.
4.
Srinivasan
,
V.
, 1993, “
The Role of Sweeps in Tolerancing Semantics
,”
Manuf. Rev.
0896-1611,
6
(
4
), pp.
275
281
.
5.
Suresh
,
K.
, and
Voelcker
,
H. B.
, 1994, “
New Challenges in Dimensional Metrology: A Case Study Based on ‘Size’
,”
Manuf. Rev.
0896-1611,
7
(
4
), pp.
291
303
.
6.
Turek
,
S.
, and
Anand
,
S.
, 2007, “
A Hull Normal Approach for Determining the Size of Cylindrical Features
,”
ASME
, Atlanta, GA.
7.
Hopp
,
T. H.
, 1993, “
Computational Metrology
,”
Manuf. Rev.
0896-1611,
6
(
4
), pp.
295
304
.
8.
Nassef
,
A. O.
, and
ElMaraghy
,
H. A.
, 1999, “
Determination of Best Objective Function for Evaluating Geometric Deviations
,”
0268-3768,
15
, pp.
90
95
.
9.
Zhang
,
X.
, and
Roy
,
U.
, 1993, “
Criteria for Establishing Datums in Manufactured Parts
,”
J. Manuf. Syst.
0278-6125,
12
(
1
), pp.
36
50
.
10.
Ramaswami
,
H.
,
Kovvur
,
Y.
, and
Anand
,
S.
, 2006, “
Accurate Size Evaluation of Cylindrical Components Using Particle Swarm Optimization
,”
Trans. North Am. Manuf. Res. Inst. SME
1047-3025,
34
, pp.
229
236
.
11.
Ramaswami
,
H.
, and
Anand
,
S.
, 2009, “
Accurate Size Evaluation of Cylindrical Components
,”
0268-3768,
49
pp.
1079
1092
.
12.
Traband
,
M. T.
,
Medeiros
,
D. J.
, and
Chandra
,
M. J.
, 2004, “
A Statistical Approach to Tolerance Evaluation for Circles and Cylinders
,”
IIE Trans.
0740-817X,
36
, pp.
777
785
.
13.
Sun
,
A. Y. T.
,
Anand
,
S.
, and
Tang
,
J. S. Y.
, 2002, “
Comprehensive Design of Experiments-Based Framework for Optimal CMM Inspection and Uncertainty Analysis of Form Tolerances
,”
Int. J. Prod. Res.
0020-7543,
40
(
9
), pp.
2097
2123
.
14.
Dowling
,
M. M.
,
Griffin
,
P. M.
,
Tsui
,
K.
, and
Zhou
,
C.
, 1995, “
A Comparison of the Orthogonal Least Squares and Minimum Enclosing Zone Methods for Form Error Estimation
,”
Manuf. Rev.
0896-1611,
8
, pp.
120
138
.
15.
Lee
,
G.
,
Mou
,
J.
, and
Shen
,
Y.
, 1997, “
Sampling Strategy Design for Dimensional Measurement of Geometric Features using Coordinate Measuring Machine
,”
Int. J. Mach. Tools Manuf.
0890-6955,
37
(
7
), pp.
917
934
.
16.
Wang
,
Z.
, 2000, “
Modeling and Sampling of Work Piece Profiles for Form Error Evaluation
,” MS thesis, University of Cincinnati, Cincinnati, OH.
17.
Feng
,
S. C.
, and
Hopp
,
T. H.
, 1991, “
A Review of Current Geometric Tolerancing Theories and Inspection Data Analysis Algorithms
,” National Institute of Standards and Technology, U.S. Department of Commerce, Technical Report No. NISTIR-4509.
18.
,
E. A.
,
Li
,
S.
, and
Chen
,
Y.
, 1997, “
Minimum Zone Cylindricity Evaluation Using Improved Nonlinear Optimization Method (NOM)
,”
Trans. NAMRI/SME
1047-3025,
25
, pp.
287
292
.
19.
Murthy
,
T. S. R.
, and
Abdin
,
S. Z.
, 1980, “
Minimum Zone Evaluation of Surfaces
,”
Int. J. Mach. Tool Des. Res.
0020-7357,
20
, pp.
123
136
.
20.
Lagarias
,
J. C.
,
Reeds
,
J. A.
,
Wright
,
M. H.
, and
Wright
,
P. E.
, 1998, “
Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions
,”
SIAM J. Optim.
1052-6234,
9
(
1
), pp.
112
147
.
21.
Uppliappan
,
B.
,
Raja
,
J.
,
Hocken
,
R.
, and
Chen
,
K.
, 1997, “
Sampling Methods and Substitute Geometry Algorithms for Measuring Cylinders in a Coordinate Measuring Machine
,”
Trans. NAMRI/SME
1047-3025,
25
, pp.
353
358
.
22.
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, Technical Report No. DITC 221/93.
23.
Turek
,
S.
, 2006, “
Cylindrical Datum Evaluation Methods Under Maximum and Least Material Condition Specifications
,” MS thesis, University of Cincinnati, Cincinnati, OH.
24.
Turek
,
S.
, and
Anand
,
S.
, 2008, “
A Comparison of Virtual Condition Cylinder Evaluation Methods in Coordinate Metrology
,”
ASME
, New York, NY.
25.
Preparata
,
F. P.
, and
Shamos
,
M. I.
, 1985,
Computational Geometry
,
Springer-Verlag
,
New York, NY
.
26.
Barber
,
C. B.
,
Dobkin
,
D. P.
, and
Huhdanpaa
,
H. T.
, 1996, “
The Quickhull Algorithm for Convex Hulls
,”
ACM Trans. Math. Softw.
0098-3500,
22
(
4
), pp.
469
483
.
27.
Roy
,
U.
, and
Xu
,
Y.
, 1995, “
Form and Orientation Tolerance Analysis for Cylindrical Surfaces in Computer-Aided Inspection
,”
Comput. Ind.
,
26
, pp.
127
134
.
28.
Roy
,
U.
, and
Zhang
,
X.
, 1992, “
Establishment of a Pair of Concentric Circles With the Minimum Radial Separation for Assessing Roundness Error
,”
Comput.-Aided Des.
0010-4485,
24
(
3
), pp.
161
168
.
30.
Zeid
,
I.
, 2005,