 Open Access
 Total Downloads : 131
 Authors : G. Rajamohan
 Paper ID : IJERTV5IS010189
 Volume & Issue : Volume 05, Issue 01 (January 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS010189
 Published (First Online): 09012016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Approach for the Unified Evaluation of Orientation Tolerances
G. Rajamohan, Ph. D
Dept. of Manufacturing Engineering National Institute of Foundry and Forge Technology
Hatia, Ranchi – 834 003 INDIA
Abstract – Orientation tolerances are specified to control the parallel, perpendicular and other angular relationships between two adjacent features of manufactured parts. One feature acts as datum feature and the other as measured or controlled feature.
Verification of orientation tolerances require the establishment of
two parallel planes at required parallel, perpendicular or angular A

A
30o 45
Datum A
0.020
30o
orientation with respect to the datum feature and encompassing
all the data points of measured feature with minimum spacing. In this paper, a new approach for unified evaluation of orientation tolerances of straight line features is proposed first followed by its implementation using the random walk, simplex search and normal least squares methods. The relative performances of these methods have been studied using simulated data.
Keywords – Angularity; Datum Feature; Measured Feature; Normal Least Squares Method; Orientation Tolerances; Parallelism; Perpendicularity; Random Walk; Simplex Search. B
12
60 25
(a) Angularity specification (b) Interpretation
0.020
B
0.020
Datum B

INTRODUCTION
The orientation tolerances are geometric tolerances that are specified with one or more datums. They are used to control the parallel (parallelism), perpendicular (perpendicularity) and
30o 45
12
60 25
90o
other angular relationships (angularity) between two adjacent features in manufactured parts. Parallelism can be defined as the condition of a surface, median plane or axis being parallel to a datum plane or axis. Perpendicularity can be defined as the condition of a surface, median plane or axis being at right angle
(c) Perpendicularity specification (d) Interpretation
0.020
C
0.020
C
Datum C
to a datum plane or axis. Angularity refers to the condition of a C surface, median plane or axis being at some specified angle to a
datum plane or axis. In all cases, the tolerance zone is defined by two parallel planes established at 0Â° for parallelism, 90Â° for perpendicularity and specified angle for angularity with respect to datum plane or axis. The elements of measured or controlled
25 45
60 25
0.020
12 12
feature must lie between these planes [1]. Fig. 1(ab), Fig. 1(c
d) and Fig. 1(ef) respectively show the example specifications of angularity, perpendicularity and parallelism tolerances and their interpretations.
Verification of orientation tolerances on measured feature requires the establishment of an ideal datum feature, based on the measurement data of datum feature, using suitable methods. The standards recommend minimum zone evaluation but do not suggest any specific method for finding the minimum zone. Despite not guaranteeing the minimum zone solution, the least squares method (LSM) is commonly used for this purpose due to its sound mathematical basis [24]. Numerous algorithms for finding the minimum zone solutions have been reported. Such algorithms are based on some optimization, soft computing and geometrybased computational techniques. A review of some of these works is presented here.
(e) Parallelism specification (f) Interpretation
Fig. 1. Examples of orientation tolerances
Computation of minimum zone form errors using numerical methods such as the MonteCarlo method, discrete Chebyshev approximation, minmax approximation, simplex search, spiral search, median technique, etc. [411], enclosing polygon based methods such as the convex hull method, Eigenpolyhedral method, etc. [10, 1214] and other methods such as control line rotation scheme [2], nonlinear optimization approach [34] and linearizing nonlinear problems using combined coordinate and scaling transformations [15] have been attempted. Use of soft computing tools, such as Genetic Algorithms (GAs) have been shown to be robust in form tolerance evaluation, e.g. circularity evaluation [16]. Computational geometric techniques were also developed for dealing with datum related features [17]. Least squares method based evaluation of the geometric tolerances in relationships, viz. parallelism, runout and concentricity, has been reported [18]. Extension of straightness evaluation using
convex hull based approach to perpendicularity evaluation has been reported [19]. An improved Particle Swarm Optimization (PSO) algorithm has been used to evaluate the perpendicularity error between two planar lines by formulating it as a linear optimization problem [20]. The authors have used maximum absolute distance as straightness error on datum feature, which violates the standards. A random walk based algorithm for the
angle for angularity (Fig. 2), 90Â° for perpendicularity (Fig. 3) and 0Â° for parallelism (Fig. 4).
Y Measured feature
t
perpendicularity evaluation has also been reported [21].
There is still a need for developing effective algorithms for the evaluation of orientation tolerances. Their evaluation also lacks a unified approach. The present work attempts to address both these issues. A unified approach is proposed first for the planar straight line features. This approach is a generalization of an earlier work meant for perpendicularity evaluation [21].
Bmax
O
Bmin
Ideal datum feature
s
X
The proposed approach is implemented using random walk, simplex search and normal least squares methods so as to find an effective algorithm. The effectiveness of these algorithms is tested using simulated data. The results obtained are presented and discussed.
Remainder of the paper is organized as follows. Section II pronounces the proposed unified approach. Section III briefs about the algorithms used for fitting the ideal datum features. Section IV presents the evaluation approach used and Section V presents and discusses the results obtained using different algorithms. Section VI states the conclusions and future scope.

THE PROPOSED UNIFIED APPROACH
As mentioned earlier, an ideal datum feature (a straight line in present case [21]) has to be established first for evaluation of orientation tolerances. Let this feature be represented as in (1), with usual notations.
Datum feature
Fig. 2. Angularity analysis ( = Specified angle)
Y
Bmax
t
Measured feature
Ideal datum feature
Datum feature
s
O
O
y ax b
(1)
Bmin X
Distances dj (j = 1, 2 m; m is the number of measurement points) between measured points Pj (xj, yj) of datum feature to the ideal datum feature may be calculated using (2).
y j ax j b
Y
Bmax
Fig. 3. Perpendicularity analysis ( = 90Â°)
Measured feature
t
d j
(1 a2 )
; j 1,2,…,m
(2)
Distance dj is taken as positive when the measured point is above the ideal datum feature and negative when i is below. Straightness error (s) of datum feature can be expressed as:
Bmin
Ideal datum
s feature
j
j
s d
max
In (3), d j

d
j
j
min
j
j
and d
min
max
(3) O
represent the minimum and maximum
X
Datum feature
values of dj respectively. Estimation of a and b for computing the straightness error (s) of datum feature can be stated as an unconstrained minimization problem, satisfying the minimum zone condition, as follows [20]:
Fig. 4. Parallelism analysis ( = 0Â°)
Lines are drawn through all the points PJ (xJ, yJ) (J = 1, 2
M; M is the number of points) of the measured feature. Let the equations of such lines be described using (6).
Minimize f (a, b) = s (4)
Y AX B
(6)
If (a*, b*) is the optimal solution for (4), equation of ideal datum feature can be written as:
In (6), A denotes the slopes and B denotes the yintercepts
of those lines. If Bmax and Bmin are the maximum (i.e. largest Y
y a*x b*
(5)
intercept) and minimum (i.e. smallest Yintercept) values of B
respectively, is the angle between ideal datum feature and the
Depending on the orientation tolerance to be evaluated, two parallel lines are established at specified angular orientation, with respect to ideal datum feature, which will be the specified
xaxis and is the specified angular orientation of measured or controlled features in relation to the datum feature, orientation tolerance (t) can be expressed as:
t (Bmax Bmin)cos( )
(7)
Start
The use of Bmax and Bmin in (7) ensures that all the points of measured feature lies within the tolerance zone. The tolerance value (t) computed using (7) is therefore the minimum value, which implies that orientation tolerances obtained using this equation follows the minimum zone evaluation.


ALGORITHMS FOR FITTING THE IDEAL DATUM FEATURE


Random Walk Method
In random walk, a sequence of improved approximations to the minimum value are generated based on the preceding approximation. If Xi is the approximation to minimum value obtained in (i1)th iteration, improved approximation in the ith iteration is obtainable from (8).
Establish the ideal datum feature using the selected algorithm
Establish the ideal datum feature using the selected algorithm
Draw lines through all points on the measured feature at specified angular orientation to the ideal datum feature
Draw lines through all points on the measured feature at specified angular orientation to the ideal datum feature
Compute the orientation tolerance
Compute the orientation tolerance
Stop
Fig. 5. Flowchart for evaluation of orientation tolerances (modified from [21])
Xi 1 Xi ui
(8)
V. RESULTS AND DISCUSSION
where, is a prescribed scalar step length and ui is a unit random vector generated in ith stage. The detailed procedure of random walk method may be found in [22].

Simplex Search Method
Simplex is a geometric figure formed by (n + 1) points in an ndimensional space. The simplex is gradually moved towards the optimal point by comparing the objective function values at (n + 1) vertices of the general simplex. Three operations, viz. reflection, contraction and expansion, are performed to achieve the desired movement of the simplex. More details of simplex search method and its algorithm may be found in [22].

Normal Least Squares Method
The objective of normal LSM is to obtain the best fit line (feature), as given in (1), that minimizes the sum of squares of normal distances (EN), between the measurement points Pj (xj, yj) and the best fit line (feature), as given in (9) [23].
Orientation tolerances refer to the geometric tolerances that are specified with datums to control the geometric deviations in manufactured parts. The verification of orientation tolerances, viz. angularity, perpendicularity and parallelism, is considered in the present work due to their importance in machine tools and coordinate measuring machines. A new unified approach for the evaluation of orientation tolerances in planar straight line features is presented in this paper. This approach has been implemented using random walk, simplex search and normal least squares algorithms using C++ language under the Visual Studio 2015 environment and run on a Microsoft Windows 10 powered PC equipped with 4 GB RAM. Performance of these algorithms were studied using the simulated datasets shown in Appendices A to C. The results of evaluation of angularity, perpendicularity and parallelism tolerances are shown in Table I, II and III respectively. As the desired criteria, evaluation algorithms are expected to yield smaller straightness values on the datum feature and preferably smaller values of orientation tolerances on the measured feature. The preferably adjective is
m
m
EN
[( y j y) cos ](9)
due to the fact that these geometrical tolerances are estimated
2
2
j 1
Substituting for y = axj + b and we get
cos
1
(1 a)2
in (9),
in sequence and not simultaneously. Precisely, smaller values of straightness error does not necessarily mean smaller values of orientation tolerances.
Table I reveals that the most commonly used normal least squares method overestimates the straightness (7.009 Âµm for A
EN
EN
( y ax b)2
and 8.944 Âµm for B) and angularity (16.120 Âµm for A and
m
j 1
j j
(1 a2 )
(10)
16.323 Âµm for B) errors. Random walk based algorithm yields smallest values of straightness (5.502 Âµm for A and 5.287 Âµm for B) and angularity (14.311 Âµm for A and 15.346 Âµm for B)
The details on normal least squares method and the method
of solving the coefficients a and b can be found in [23].
IV. APPROACH TO COMPUTATION OF ORIENTATION TOLERANCES
Flowchart for the approach used to evaluate the orientation tolerances is shown in Fig. 5. The random walk, simplex search and normal least squares method based algorithms are used only for obtaining the ideal datum feature. After this, lines are drawn through all points of measured or controller feature at specified angular orientation with reference to the ideal datum feature. If Bmax is the yintercept of topmost line and Bmin is the yintercept of bottommost line, the orientation tolerance (t) can be computed by substituting these values in (7).
errors and simplex search comes next. Table II reveals similar trends in perpendicularity evaluation. The results of parallelism error evaluation, shown in Table III, reveals that random walk algorithm outperforms the other algorithms, i.e. yields lowest straightness and parallelism errors. The simplex search shows a slightly poor performance in parallelism error in comparison to normal LSM, however, the straightness error is still less than that of normal LSM. Thus, the random walk based algorithm is found to perform consistently in evaluating the orientation tolerances better than the other two algorithms. As a general statement, it may be said that normal LSM overestimates the geometrical tolerances always despite its sound mathematical basis and wide application in measuring instruments. Random walk based algorithm also follows the standards.
15 
10.50 
2.5165 
8.2044 
7.0261 
15 
14.00 
2.5175 
4.0992 
13.6239 
16 
11.25 
2.5161 
8.8736 
7.3648 
16 
15.00 
2.5163 
4.5562 
14.5134 
p>17 
12.00 
2.5161 
9.5428 
7.7035 
17 
16.00 
2.5154 
5.0132 
15.4029 
18 
12.75 
2.5153 
10.2112 
8.0436 
18 
17.00 
2.5138 
5.4688 
16.2930 
19 
13.50 
2.5131 
10.8804 
8.3823 
19 
18.00 
2.5106 
5.9259 
17.1825 
20 
14.25 
2.5145 
11.5477 
8.7246 
20 
19.00 
2.5111 
6.3794 
18.0737 
21 
15.00 
2.5123 
12.2176 
9.0620 

22 
15.75 
2.5111 
12.8866 
9.4009 

23 
16.50 
2.5105 
13.5553 
9.7407 

24 
17.25 
2.5090 
14.2242 
10.0798 

25 
18.00 
2.5093 
14.8948 
10.4157 
15 
10.50 
2.5165 
8.2044 
7.0261 
15 
14.00 
2.5175 
4.0992 
13.6239 
16 
11.25 
2.5161 
8.8736 
7.3648 
16 
15.00 
2.5163 
4.5562 
14.5134 
17 
12.00 
2.5161 
9.5428 
7.7035 
17 
16.00 
2.5154 
5.0132 
15.4029 
18 
12.75 
2.5153 
10.2112 
8.0436 
18 
17.00 
2.5138 
5.4688 
16.2930 
19 
13.50 
2.5131 
10.8804 
8.3823 
19 
18.00 
2.5106 
5.9259 
17.1825 
20 
14.25 
2.5145 
11.5477 
8.7246 
20 
19.00 
2.5111 
6.3794 
18.0737 
21 
15.00 
2.5123 
12.2176 
9.0620 

22 
15.75 
2.5111 
12.8866 
9.4009 

23 
16.50 
2.5105 
13.5553 
9.7407 

24 
17.25 
2.5090 
14.2242 
10.0798 

25 
18.00 
2.5093 
14.8948 
10.4157 
TABLE I. RESULTS OF ANGULARITY EVALUATION
Dataset 
Algorithm 
Straightness error, Âµm 
Angularity error, Âµm 
A 
Random Walk 
5.502 
14.311 
Simplex Search 
6.737 
15.794 

Normal LSM 
7.009 
16.120 

B 
Random Walk 
5.287 
15.346 
Simplex Search 
8.668 
16.020 

Normal LSM 
8.944 
16.323 
TABLE II. RESULTS OF PERPENDICULARITY EVALUATION
Dataset 
Algorithm 
Straightness error, Âµm 
Perpendicularity error, Âµm 
C 
Random Walk 
5.889 
12.590 
Simplex Search 
7.701 
14.752 

Normal LSM 
7.849 
14.930 

D 
Random Walk 
4.711 
19.300 
Simplex Search 
7.893 
22.854 

Normal LSM 
8.132 
23.116 
TABLE III. RESULTS OF PARALLELISM EVALUATION
Dataset 
Algorithm 
Straightness error, Âµm 
Parallelism error, Âµm 
E 
Random Walk 
6.449 
25.824 
Simplex Search 
9.677 
31.090 

Normal LSM 
10.028 
30.537 

F 
Random Walk 
5.316 
17.909 
Simplex Search 
10.643 
21.473 

Normal LSM 
10.866 
21.166 
VI. CONCLUSIONS AND FUTURE SCOPE Geometrical deviations in manufactured parts are caused by
systematic and random errors that occur during manufacturing. The geometrical tolerances are used to specify and control such deviations. Their evaluation requires effective algorithms that follow the standards. The evaluation of orientation tolerances, viz. angularity, perpendicularity and parallelism in straight line features has been considered in the present work. An approach for unified evaluation orientation tolerances has been proposed
APPENDIX B: PERPENDICULARITY DATASET
S# 
Dataset C 
S# 
Dataset D 

Datum 
Measured 
Datum 
Measured 

xj 
yj 
xj 
yj 
xj 
yj 
xj 
yj 

1 
0.00 
2.5158 
2.5244 
0.00 
1 
0.00 
2.5122 
2.5292 
0.00 
2 
0.80 
2.5159 
2.5261 
0.80 
2 
1.20 
2.5124 
2.5318 
1.20 
3 
1.60 
2.5157 
2.5289 
1.60 
3 
2.40 
2.5122 
2.5353 
2.40 
4 
2.40 
2.5158 
2.5290 
2.40 
4 
3.60 
2.5122 
2.5359 
3.60 
5 
3.20 
2.5183 
2.5288 
3.20 
5 
4.80 
2.5146 
2.5358 
4.80 
6 
4.00 
2.5181 
2.5315 
4.00 
6 
6.00 
2.5141 
2.5384 
6.00 
7 
4.80 
2.5191 
2.5285 
4.80 
7 
7.20 
2.5148 
2.5350 
7.20 
8 
5.60 
2.5164 
2.5321 
5.60 
8 
8.40 
2.5118 
2.5380 
8.40 
9 
6.40 
2.5206 
2.5317 
6.40 
9 
9.60 
2.5156 
2.5366 
9.60 
10 
7.20 
2.5189 
2.5297 
7.20 
10 
10.80 
2.5134 
2.5334 
10.80 
11 
8.00 
2.5173 
2.5309 
8.00 
11 
12.00 
2.5112 
2.5332 
12.00 
12 
8.80 
2.5183 
2.5303 
8.80 
12 
13.20 
2.5116 
2.5310 
13.20 
13 
9.60 
2.5160 
2.5308 
9.60 
13 
14.40 
2.5086 
2.5297 
14.40 
14 
10.40 
2.5193 
2.5294 
10.40 
14 
15.60 
2.5112 
2.5264 
15.60 
15 
11.20 
2.5175 
2.5292 
11.20 
15 
16.80 
2.5087 
2.5241 
16.80 
16 
12.00 
2.5169 
2.5270 
12.00 
16 
18.00 
2.5075 
2.5198 
18.00 
17 
12.80 
2.5167 
2.5250 
12.80 
17 
19.20 
2.5066 
2.5156 
19.20 
18 
13.60 
2.5158 
2.5246 
13.60 
18 
20.40 
2.5051 
2.5130 
20.40 
19 
14.40 
2.5134 
2.5226 
14.40 
19 
21.60 
2.5021 
2.5090 
21.60 
20 
15.20 
2.5146 
2.5246 
15.20 
20 
22.80 
2.5028 
2.5090 
22.80 
21 
16.00 
2.5123 
2.5211 
16.00 

22 
16.80 
2.5109 
2.5194 
16.80 

23 
17.60 
2.5101 
2.5186 
17.60 

24 
18.40 
2.5083 
2.5172 
18.40 

25 
19.20 
2.5085 
2.5122 
19.20 
APPENDIX C: PARALLELISM DATASET
Dataset E Dataset F
S# S#
and implemented using different numerical algorithms, viz. random walk, simplex search and normal least squares method. The performance of these algorithms has been evaluated using simulated data. Random walk algorithm has been found to be most effective among the three algorithms. The simplex search algorithm has also been found to yield better results. Both these algorithms outperform the most commonly used normal least squares based algorithm. The extension of proposed approach to threedimensional features may form the future work.
APPENDIX A: ANGULARITY DATASET
Datum Measured
xj yj xj yj
2 
1.00 
2.5161 
1.00 
47.5366 
2 
1.25 
2.5176 
1.25 
77.5306 
3 
2.00 
2.5160 
2.00 
47.5402 
3 
2.50 
2.5177 
2.50 
77.5341 
4 
3.00 
2.5162 
3.00 
47.5408 
4 
3.75 
2.5180 
3.75 
77.5346 
5 
4.00 
2.5187 
4.00 
47.5410 
5 
5.00 
2.5205 
5.00 
77.5345 
6 
5.00 
2.5184 
5.00 
47.5439 
6 
6.25 
2.5201 
6.25 
77.5370 
7 
6.00 
2.5193 
6.00 
47.5443 
7 
7.50 
2.5208 
7.50 
77.5335 
8 
7.00 
2.5165 
7.00 
47.5409 
8 
8.75 
2.5176 
8.75 
77.5363 
9 
8.00 
2.5205 
8.00 
47.5435 
9 
10.00 
2.5212 
10.00 
77.5348 
10 
9.00 
2.5185 
9.00 
47.5410 
10 
11.25 
2.5187 
11.25 
77.5315 
11 
10.00 
2.5166 
10.00 
47.5414 
11 
12.50 
2.5162 
12.50 
77.5311 
12 
11.00 
2.5172 
11.00 
47.5400 
12 
13.75 
2.5162 
13.75 
77.5288 
13 
12.00 
2.5145 
12.00 
47.5394 
13 
15.00 
2.5127 
15.00 
77.5273 
14 
13.00 
2.5173 
13.00 
47.5368 
14 
16.25 
2.5147 
16.25 
77.5238 
15 
14.00 
2.5150 
14.00 
47.5352 
15 
17.50 
2.5117 
17.50 
77.5214 
16 
15.00 
2.5140 
15.00 
47.5316 
16 
18.75 
2.5098 
18.75 
77.5170 
17 
16.00 
2.5133 
16.00 
47.5281 
17 
20.00 
2.5083 
20.00 
77.5127 
18 
17.00 
2.5118 
17.00 
47.5260 
18 
21.25 
2.5062 
21.25 
77.5101 
19 
18.00 
2.5088 
18.00 
47.5224 
19 
22.50 
2.5026 
22.50 
77.5060 
20 
19.00 
2.5096 
19.00 
47.5227 
20 
23.75 
2.5027 
23.75 
77.5061 
21 
20.00 
2.5067 
20.00 
47.5176 

22 
21.00 
2.5048 
21.00 
47.5143 

23 
22.00 
2.5036 
22.00 
47.5120 

24 
23.00 
2.5015 
23.00 
47.5090 

25 
24.00 
2.5013 
24.00 
47.5027 
2 
1.00 
2.5161 
1.00 
47.5366 
2 
1.25 
2.5176 
1.25 
77.5306 
3 
2.00 
2.5160 
2.00 
47.5402 
3 
2.50 
2.5177 
2.50 
77.5341 
4 
3.00 
2.5162 
3.00 
47.5408 
4 
3.75 
2.5180 
3.75 
77.5346 
5 
4.00 
2.5187 
4.00 
47.5410 
5 
5.00 
2.5205 
5.00 
77.5345 
6 
5.00 
2.5184 
5.00 
47.5439 
6 
6.25 
2.5201 
6.25 
77.5370 
7 
6.00 
2.5193 
6.00 
47.5443 
7 
7.50 
2.5208 
7.50 
77.5335 
8 
7.00 
2.5165 
7.00 
47.5409 
8 
8.75 
2.5176 
8.75 
77.5363 
9 
8.00 
2.5205 
8.00 
47.5435 
9 
10.00 
2.5212 
10.00 
77.5348 
10 
9.00 
2.5185 
9.00 
47.5410 
10 
11.25 
2.5187 
11.25 
77.5315 
11 
10.00 
2.5166 
10.00 
47.5414 
11 
12.50 
2.5162 
12.50 
77.5311 
12 
11.00 
2.5172 
11.00 
47.5400 
12 
13.75 
2.5162 
13.75 
77.5288 
13 
12.00 
2.5145 
12.00 
47.5394 
13 
15.00 
2.5127 
15.00 
77.5273 
14 
13.00 
2.5173 
13.00 
47.5368 
14 
16.25 
2.5147 
16.25 
77.5238 
15 
14.00 
2.5150 
14.00 
47.5352 
15 
17.50 
2.5117 
17.50 
77.5214 
16 
15.00 
2.5140 
15.00 
47.5316 
16 
18.75 
2.5098 
18.75 
77.5170 
17 
16.00 
2.5133 
16.00 
47.5281 
17 
20.00 
2.5083 
20.00 
77.5127 
18 
17.00 
2.5118 
17.00 
47.5260 
18 
21.25 
2.5062 
21.25 
77.5101 
19 
18.00 
2.5088 
18.00 
47.5224 
19 
22.50 
2.5026 
22.50 
77.5060 
20 
19.00 
2.5096 
19.00 
47.5227 
20 
23.75 
2.5027 
23.75 
77.5061 
21 
20.00 
2.5067 
20.00 
47.5176 

22 
21.00 
2.5048 
21.00 
47.5143 

23 
22.00 
2.5036 
22.00 
47.5120 

24 
23.00 
2.5015 
23.00 
47.5090 

25 
24.00 
2.5013 
24.00 
47.5027 
1 0.00 2.5158 0.00 47.5341 1
Datum Measured
xj yj xj yj
0.00 2.5170 0.00 77.5280
S# 
Dataset A 
S# 
Dataset B 

Datum 
Measured 
Datum 
Measured 

xj 
yj 
xj 
yj 
xj 
yj 
xj 
yj 

1 
0.00 
2.5146 
1.1482 
2.2536 
1 
0.00 
2.5182 
2.2557 
1.1493 
2 
0.75 
2.5146 
0.4808 
2.5957 
2 
1.00 
2.5187 
1.8038 
2.0414 
3 
1.50 
2.5143 
0.1861 
2.9388 
3 
2.00 
2.5187 
1.3529 
2.9340 
4 
2.25 
2.5144 
0.8542 
3.2795 
4 
3.00 
2.5190 
0.8994 
3.8253 
5 
3.00 
2.5168 
1.5225 
3.6200 
5 
4.00 
2.5216 
0.4455 
4.7163 
6 
3.75 
2.5165 
2.1895 
3.9630 
6 
5.00 
2.5214 
0.0059 
5.6086 
7 
4.50 
2.5175 
2.8590 
4.3009 
7 
6.00 
2.5223 
0.4627 
6.4982 
8 
5.25 
2.5149 
3.5256 
4.6447 
8 
7.00 
2.5195 
0.9136 
7.3908 
9 
6.00 
2.5191 
4.1940 
4.9849 
9 
8.00 
2.5235 
1.3683 
8.2814 
10 
6.75 
2.5174 
4.8631 
5.3237 
10 
9.00 
2.5215 
1.8245 
9.1713 
11 
7.50 
2.5159 
5.5308 
5.6654 
11 
10.00 
2.5195 
2.2781 
10.0625 
12 
8.25 
2.5170 
6.1993 
6.0055 
12 
11.00 
2.5201 
2.7333 
10.9529 
13 
9.00 
2.5148 
6.8672 
6.3465 
13 
12.00 
2.5173 
3.1877 
11.8437 
14 
9.75 
2.5182 
7.5361 
6.6858 
14 
13.00 
2.5199 
3.6439 
12.7336 
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BIOGRAPHY
G. Rajamohan received his B. E. (Mechanical Engg.) from College of Engineering, Anna University, Chennai, M. E. (Manufacturing Tech.) from Regional Engineering College, Tiruchirappalli and Ph. D. (Mechanical Engg.) from Indian Institute of Technology Madras. He is associated with the
National Institute of Foundry & Forge Technology, Ranchi since 1998. His research interests include metrology and computer aided inspection, machining, image processing and applications of computers in design and manufacturing.