establishing reference figures for form evaluation of engineering surfaces

Establishing Reference Figures for Form Evaluation of Engineering Surfaces

M.S. Shunmugam, Michigan Technological University, Houghton, MI

Abstract A generalized algorithm is presented in this paper to

establish the reference figures as specified in the international and national standards on form errors. This algorithm is guaranteed to give optimal results and is well suited for implementation on a computer- interfaced, form-measuring instrument or a coordinate measuring machine.

Keywords: Metrology, Engineering Surfaces, Form Error, Reference Figures, Algorithm.

Introduction Dimensional and geometr ic tolerances are

assigned to selected features of engineering components to satisfy certain functional or assembly requirements. Since every manufacturing operation is associated with systematic and random variations, verification of the manufactured components is done through manual checking against standard gages or by appropriate measurements. The introduction of coordinate measuring machines (CMMs) has facilitated the measurements greatly. This also provides greater integration flexibility with CAD systems. Therefore, there is a growing interest among researchers to develop suitable methods for the evaluation of geometric errors with a particular emphasis on the evaluation of form errors.

According to ISO, the form of a single toleranced feature is deemed to be correct when the distance of its individual points from a superimposed surface of ideal geometric form is equal to or less than the value of the specified tolerance. ~ The orientation of the ideal geometric form should be chosen so that the distance between it and the actual surface of the feature concerned is the least possible value. Many other national standards follow the above definition.

A few standards also mention the use of minimum circumscribing and maximum inscribing circles for the evaluation of roundness error. But none of these standards specify the methods to establish ideal geometric features. 2' 3

Earlier attempts were made to establish geometric features based on the least squares technique and are also incorporated in some standards. 2' 3 The value of the form error obtained by this method will not be the minimum. This may lead to rejection of parts that are actually within the tolerance range. This technique can also lead to a certain degree of disagreement between the measurement and gaging results (for example, ring and plug gaging). There- fore, many attempts have been made to develop methods to arrive at the ideal geometric figures as outlined in the standards.

Murthy and Abdin discussed the use of Monte Carlo, Simplex, and spiral search techniques for the evaluation of circularity and flatness according to the min imum zone principle, a This approach requires extensive computer time. Fukuda and Shimokobhe used a minimax approximation to obtain this solution. 9 They claimed that the time required using their algorithm for a large number of points would be less than that using the least squares technique. But it is not clear whether the time required for replacement of points has been included. Obviously, a search procedure will require more time than a one-step procedure such as the least squares technique. Also, the method used for replacement of points has not been fully described. Chetwynd formulated this minimum zone approach as a problem in linear programming and extended it to cover the minimum circumscribed and maximum inscribed circles. He sug- gested a 180 ° rule for the enveloping circles that stated that no two adjacent points out of the three

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Journal of Manufacturing Systems Volume 10/No. 4

chosen data points subtend an angle at the origin of more than 180 ° . This work deals mainly with roundness analysis and very little with flatness analysis. 6

This paper presents a general algorithm that establishes the reference figures as outlined in the standards. Every attempt has been made to over- come problems that occur with other algorithms. This paper covers typical geometrical features such as straight line, plane, circle, and cylinder. The min imum zone figure is established on the basis of the theory of discrete and linear Chebyshev approx imat ion . H The general a lgor i thm also extends to cover the enveloping figures that are the equivalent of the min imum circumscribed and maximum inscribed circles. The criterion for optimality is also described in each case. Simple numerical examples that serve as test data are also included.

Definition of Problem Let f be a function representing the actual sur-

face, the values of which are obtained as a set of symmetrical data points by measurements. We have to find a function ~b that fits the actual surface according to different criteria. The ideal form of most of the figures like straight line, circle, etc., can be expressed in the form of linear functions, s, 7 This is valid as long as the measured feature is well aligned with the measurement datum. 4 This condition is satisfied in many form-measuring instruments by taking proper care during the setting phase. However , the data from a CMM must be suitably transformed as explained by the author elsewhere. ~z Hence, we can express + as

+i : a l Uil ~- a2 bli2 "~-

. . . . . + a j uij q- a m Uim ( l )

where m is the number of variables, uij denotes the value of the j th variable at the ith point and aj the coefficient of the j th variable.

The error at the point i is expressed as

ei : ¢ b i - f i (2)

The form error is computed as

h t : lemaxl q - l e m i n l (3)

where ema x and emi n are the max imum and mini- m u m errors, respectively.

In the present work, given u~/and f/, we must find af t of ~b so that certain criterion is satisfied. In the method of least squares, we try to find ~b so that

E [ei] z is a min imum (4)

The min imum zone evaluation of surfaces as described by ISO, ~ requires that the max imum value of the absolute error is minimized, i.e.

{maximum )e i I} is a min imum (5)

The enveloping crest figure (equivalent of min imum circumscribing circle/cylinder) has all the measured points within it and the enclosed area/volume at its min imum value, i.e.

e i >-- 0 and a~ is a min imum (6)

Similarly, the enveloping valley figure (equivalent of max imum inscribed circle/cylinder) has all the measured points outside of it and the area/volume enclosed is the maximum. This will result in

e i <-- 0 and a 1 is a max imum (7)

General Algorithm Here, we describe the algorithm in its most

general form. The flowchart of the algorithm is shown in F i g u r e 1.

S tep 1: Choose a set of m points for the enveloping figures and (m + 1) points for the min imum zone figure to serve as an initial reference set. Although any set of points may be chosen, it is recommended that the extreme points with reference to the least squares figure are taken. This not only avoids ambiguity, but also results in a reduced number of trials. Represent these points by vkj such that vtq = u o and i take the value of points in the reference set.

S tep 2: Using the reference set, solve for af t .

m

Z a j vkj j = l

= fk + (sgn([3k)) H (8)

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Journal of Manufacturing Systems V o l u m e 10/No. 4

CTz-) ! / Read Data /

I Choose reference figure

. I I C oose a reference set [ ~

I i~;~'~~ Y;~~~i~;~i ira+ l p o~n,~ /

- -~ fh~61 ih=oi :

No I Calct [ate H

Solve for ajs in ~ajvkj = fk + (sgn(l~k))H

Calculate e i

Check for optimality ...... Is . . . . . . . . . . . . . . . . . i~' . . . . . . . . . is ........

. . . . . . . . . . * < , i .q Yes a I rain? .. a I m a x ? : . . . . ::ii ] e [

....... ei_>0 ..." ..... .. - . "..... e <0 ..... . . . . ....

No N i Yes ..... " ....... ;:/ No

By trial ! ] By trial i . . . . . . . . . . . . . . . . . . . . . . . . . ~,g~)h~!:,:.,.,: ~,~

~Find erron. Ill. fonu 1

Figure 1 Flow Chart of the Form Evaluation Algorithm

a) For enveloping figures, k takes the value of 1 to m and the reference deviation H is taken to be zero.

b) Though k takes the value of I to m + 1, we have to solve only m equations for the minimum zone figure, due to the following characteristic rela- tion.

m + l

j = l

= 0 (9)

The general solution for f~kS is expressed as a determinant

13 k = ( -1 ) k+llvijl j = 1 . . . . . . . . m (10) j = 1, , m + l (j:/:k)

The reference deviation H is given by

m + l m + l

k = l k = l

(If any [3 k = 0, Haar's condition is violated and positive and negative signs must be tried for H in Equation 8. If EII3kl = 0 leading to a degenerate condition, a new set must be chosen)

Step 3: Find the error e i of the reference function at all given points and the form error is calculated based on Equation 3. From the e i s, the prime error e* is found.

a) The prime error is taken to be the least error in case of the crest figure and the highest value for the valley figure, leaving out the errors at the reference points.

b) For minimum zone approach, the value of e i whose absolute value is the highest is considered to be the prime error.

Step 4: Check for optimality

a) Each point in the reference set is replaced in turn by the prime error point. The value of a I and the error at the omitted point e' are computed. For the crest figure, the search is continued if a I is less than the other values of a~ and e' is positive. A greater value of a~ and a negative e' lead to another search in case of valley figure.

b) If le*[<--H[,the criterion of optimality is satisfied and the search is stopped. Otherwise, go to Step 5.

Step 5: Include the point i* corresponding to the prime error e m the reference set. The point to be discarded from the reference set is found in the following ways.

a) While checking for optimality, the set giving a positive e' and a minimum a I is identified and used in the 'trial in establishing a crest figure. In case of a valley figure, this would be a set giving a negative e' and a maximum a~.

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Journal of Manufacturing Systems V o l u m e 10/No. 4

b) For minimum zone approach, if 13k~:0 the Stiefel Exchange theorem l° can be used. Assum- ing a suitable value for one of the ixks, the following equation is solved.

m + l

~ , ~l, kVkj Jr Ui. j k : l

= 0 j : 1 . . . . . . . . m (12)

Calculate the value of qk

qk = Ixk/kk k = 1 . . . . . . . m + 1 (13)

If [sgn(H)e*] > 0, then the point with a minimum qk should be removed from the set. Other- wise, the point with maximum qk is discarded.

If any [31, -- 0, we must resort to Trial Exchange--replacing each point one at a time and finding the combination that leads to the largest value of IHI.

With this new reference set, go to Step 2.

for the crest figure. In this case, the points 1 and 2 are taken as the reference set. The prime error occurred at point 3 in the first trial. This point along

Table lb Straightness Error--Least Squares Line

1

a 2

1 2 3 4

5

2 . 6

- 0 . 6

3 . 8 0 . 8 3 . 2 - 1 . 8 2 . 6 0 . 6 2 . 0 1 . 0 1 . 4 - 0 . 6

h 2.8 t

Table l c Straightness Error--Crest Line

T r i a l 1 T r i a l 2 T r i a l 3

k 1 fk 1 fk 1 fk

Applications Typical geometrical features are considered in

this section and the results are presented in Tables 1-4.

Straightness: The data from the straightness measurement are denoted by {x i, f/} as in Figure 2. The approximation line is given as

~ i = a l + a2 xi ( 1 4 )

a 1 represents the intercept and a 2 the slope. Com- pared with Equation 1, it is seen that uil = 1 and uiz = X i.

Table la gives the measurement data. Results of the least squares approximation are included in Table lb. Table lc deals with the results obtained

Table la Straightness Error--Measurement Data: f,. Ix m

x I - 2 -1 0 1 2

f i ( l } 3 (1 ) 5 (2 ) 2 (3 ) 1(4) 2 (5 )

1 2

a 1

a 2

i

1 2 3 4 5

h t

e I l

1 3 2 5 2 5 2 5 3 2 5 2

7 .0 2 .0 4 .0

2 .0 - 3 . 0 - 1 . 0

~l et Ct e ~l e

3.0 0.0 8.0 5.0 6.0 3.0 5 .0 0 . 0 5 .0 0 .0 5 .0 0 .0 7 .0 5 .0 2 . 0 0 . 0 4 , 0 2 . 0 9 .0 8 .0 - 1 . 0 - 2 . 0 3 ,0 2 . 0

11.0 9 .0 - 4 . 0 - 6 . 0 2 ,0 0 .0

9 .0 11.0 3 .0

5 .0 - 6 . 0 3 .0 3 5 1

l fk i fk 1 fk t fk 1 fk 1 fk

1 3 2 5 2 5 3 2 2 5 3 2 3 2 3 2 5 2 5 2 4 1 5 2

2 .0 2 ,0 4 .0 2 .0 3 .0 2 .0

- 2 . 5 S,O 2 .0 - 3 . 0 - 4 . 0 - 3 . 0 {e 2) (e t } (e 3 } (e 2 } (e S ) (e 2)

317


Table I d Straightness Error--Valley Line

k

1 2

8.1

8" 2

Tris.1 1 Tris.1 2 Tr ia l 3

1 fk 1 fk i fk

1 3 1 3 1 3 2 5 5 2 4 1

7 . 0 2 . 5 1 . 6 6

2 . 0 - 0 . 2 5 - 0 . 6 6

1 ~ el @l e ~t et

1 3 . 0 O.O 3 . 0 0 . 0 3 . 0 O.O 2 5 . 0 O. 0 2 . 7 5 - 2 . 2 5 2 . 3 3 - 2 . 6 6 3 7 . 0 5 . 0 2 . 5 0 . 5 1 . 6 6 - 0 . 3 3 4 9 . 0 8 . 0 2 . 2 5 1 . 2 5 1 . 0 0 0 . 0 5 1 1 . 0 9 . 0 2 . 0 0 . 0 0 . 3 3 - 1 . 6 6

h t 9 . 0 3 . 5 2 . 6 6

t e 9 . 0 1 . 2 5 - 0 . 3 3 1 6 4 3

k i £k 1 £k 1 £k i £k 1 £k i £k

1 1 3 2 S 4 1 1 3 1 3 3 2 2 S 2 S 2 5 2 4 1 3 2 4 1

a t 2 . 5 4 . 0 O.O 1 . 6 6 2 . 0 2 . 0

e" 2 . 2 5 3 . 0 - B . O - 0 . 3 3 0 . 5 1 . 0

( e 2 ) (e I ) (e I ) (e S ) (e L ) (e I )

with point 2 results in a positive error e' and a value for a I less than that obtained with points 1 and 2. The next trial is continued with these two points. At the end of the third trial, we find that the subtrials lead to a negative error e ' , indicating the termina- tion of the search.

The results of the valley line are shown in Table lc . In case of the minimum zone approach, the exchange is carried out according to the Stiefel Exchange theorem as shown in Table le . Point 3 with a minimum qk is replaced by the prime error point, since [sgn(H)e*] is greater than zero in this trial.

Flatness: Figure 3 shows the flatness measurement represented by {x i, Yi~}. The approximation plane is given by

d# i = a 1 + arri + a3Y i (15)

where a 2 and a 3 represent the slopes.

Only the final results are given in Table 2.

Circularity: The circularity measurements are given by {Oi, f/} as shown in Figure 4. The approximation circle s' 7 is written as

Table l e Straightness Error--Minimum Zone Line

1

8. 2

1 2 3 4 5

h t

o e . I

k

T r i a l 1

£ k

1 1 3 1 2 3 2 - 2 5 2 - 3 5 3 1 2 4 1 1

1 . 2 5 1 . 3 3

3 . 2 5 3 . 0 0

- 0 . 5 0 - 0 . 6 6

~1 el @i et

4 . 2 5 1 . 2 5 3 . 7 5 - 1 . 2 5 3 . 2 5 1 . 2 5 2 . 7 5 1 . 7 5 2 . 2 5 0 . 2 5

3. O0

1 . 7 5 4

Pk qk

3 . 0 3 .0 - 5 . 0 2 . 5

1 . 0 1 . 0

T r i a l 2

i ~k f k

4 . 3 3 1 . 3 3 3 . 6 6 - 1 . 3 3 3 . 0 0 1 . 0 0 2 . 3 3 1 . 3 3 1 . 6 6 - 0 . 3 3

2 . 6 6

1 . 3 3 1 , 2 , 4

f~i ---- aj + a 2 cos Oi + a3 sin 0 i (16)

where a I is the radius and (az,a3) is the center. In this case, uil = 1, uiz = cos Oi and ui3 = sin O r Table 3 shows the final results.

Cylindricity: The data in the form {zi,Oi, fi} represents the measurement as in Figure 5. The approximation function is

~i = al + az cos 0 i + a 3 sin 0 i + a4z i c o s 0 i

+ as zi sin 0 i (17)

Table 4 shows the final results.

Discussion The enveloping figures such as minimum circum-

scribed circle and maximum inscribed circle are the mathematical equivalents of the ring-gage and plug-

3 1 8


gage, respectively. To establish these features, a certain geometrical condition known as the 180 ° rule has been recommended in the literature. 6 According to this, the angle subtended at the origin by any two adjacent points out of the three chosen data points should not be more than 180 °. This rule needs modification for other geometrical features; namely, straightness, flatness, and cylindricity. Otherwise, cyclical exchanges occur leading to the selection of the same reference set again and again. Therefore, a trial exchange is followed in the present method. This trial exchange procedure works for all geometrical features. The optimality condition is also unique for the enveloping figures.

Figure 2 Evaluation of Straightness Error

a) Deviation From a Straight Line b) Enveloping Crest and Valley Lines c) Minimum Zone Line

(~i = ai + a2xi

xi

a) Deviation from a straight line

1-

-1

Crest Line

t~ i = 4 . 0 0 - x i

1 2

b) Enveloping crest and valley lines

H = 1 - 1 . 3 3 1

H = [ + 1.331

-1 -1 c) M i n i m u m zone line

5 / Minhnum zone / l ine

~ .66 ×i

H = I [ 1 3 3 ' I

I 2

H

a 1

a 2

a 3

h t

F%ure 3 Deviation From a Plane

xi

°, a, aj:a,y,

Table 2 Flatness Error

a) Measurement data: f ( i ) ~m l

1

Yl o

-I

X !

-2 -1 0 1 2

5(i) 4(2) I(3) 2(4) 2(5)

4(6) 3(7) 3(8) 2(9) 2(10)

3(II) 4(12) 2(13) 1(14) 2(IS)

b) Error evaluation

Least Squares

2 . 6 6

- 0 . 6

0 . 2

Mln, Zone Crest Valley

i ~k fk I fk i fk

3 8 1 12 4 11 3 S - 4 2 1 5 3 1 1 - 4 5 15 2 14 1

11 0 3

1 .25 0 . 0 0 . 0

2 . 5

- 0 . 7 5

- 0 . 2 5

3 . 5

- 0 . 6 6

0 . 1 6

1 .33

- 0 . 6 6

- 0 . 3 3

2 . 8 2 . 5 2 . 6 6 2 . 6 6

319


F@ure 4 Deviation From a Circle

y Approxhnation / ' - ' - - Ci rc le

/ ~ i = al + a2cos 0i + a3sin 0i

X

Table 3 Circularity Error

In case of the minimum zone figure, the condition that ke*k>q H I shows that there is a better approximation function available. At each stage, IHI is increased monotonically. In the linear programming approach, we try to minimize qe* I. Here we are actually maximizing I/-/], the reference deviation. This is the difference between this algorithm and the linear programming approach. We do not bother increasing the value of le*] at some trial. Hence, this is also known as an ascent algorithm. The points are exchanged based on the Stiefel Exchange theorem. No additional geometrical conditions are necessary to avoid the recycling exchanges. However, if any [31, = 0, we must resort to trial exchanges.

To see the efficiency or fast convergence of the algorithm, consider the case of the straightness error calculation with thirty points. If an exhaustive search is done for the crest or valley line, 3°C 2 =

Figure 5 Deviation From a Cylinder

H

a 1

a 2

a 3

h t

a) Measurement data: fl ~m

e[ deg 0 4S 90 135 180 225 270 315

f i ( i ) #m 4(1) 4 (2) 3(3) S(4) 2[~) 3 (6) 1(7) 2 (8)

b) Error evaluatlon

Least Squares

3 .0

O. 14

1.2

k 1 ~k fk

Mln. Zone Crest V a l l e y

I

1 4 2 .0 S 2 3 -1 .41 3 3 5 - 1 . 0 2 4 1 0.41 4

- 1 . 1 2

1 fk

4 S 1 4 6 3

0.0

4.0

0.0

1.41

3 .0

-0, 12

1.12

3 3 S 2 7 1

0.0

2 . 0

0 . 0

1.0

2.45 2,24 2.41 2.29

fk

Approximation

Cylinder

~i = al + a2cos0i + a3sin 0i + a4zicos 0 i + aszis in 0 i

fi

Y

o a2 X

Y

o x

a 3 + aSz i

a

Oi

a 2 + a4z i

320


Table 4 Cylindricity Error

a) Measurement d a t a : f ( t ) ~m

O I deg

0 45 90 135 180 22S 270 315

5(1 ) 3 (2 ) 4(3) 3(4) 1(5) 2(6 ) 2(7) 3(8)

4 ( 9 ) 4 ( 1 0 ) 3 (11 ) 3 (12 ) 3 (13 ) 2 (14 ) 2 ( 1 5 ) 3 (16 )

3 (17 ) 4 (18 ) 3 (19 ) 2 (20 ) 3 (21 ) 1(22) 2 (23 ) 2 (24 )

b) E r r o r e v a l u a t i o n :

H

a 1 2.79

a 2 0,55

a 3 0.67

a4 0.64

a s -0.03

h t 2.24

Le&st Squares

Min. Zone Crest Valley

k i Bk fk i fk I fk

1 20 - 4 . 0 2 4 3 2 3 2 2 - 6 . 8 2 3 13 3 18 2 3 22 - 2 . 8 2 1 1 5 S 1 4 1 9 . 6 5 5 22 3 21 2 5 23 - 2 . 8 2 2 8 2 23 1 6 4 6 . 8 2 3

- 0 . 9 6 0 . 0 0 . 0

2 . 6 7

0 , 5 3

0 . 6 8

0 . 8 2

- 0 . 1 2

3, 53

O. 46

O. 95

1. O0

- 0 . 2 4

2 . 0 0

1 . 0 0

0 . 7 1

0. O0

0 . 2 9

1,92 2.76 2.00

335 combinations must be searched. Similarly, 3OC 3 = 4030 must be tried for a minimum zone figure. But with this algorithm, three exchanges are usually sufficient to obtain the optimum solution.

Conclusion A generalized algorithm has been presented that

can establish reference figures outlined in the standards. An attempt has been made to reduce mathematical complexity as far as possible to find the widest application and acceptance.

The enveloping figures serve as soft-gages and can easily replace conventional gages used for inspection. The algorithm is also used to find the minimum separation enveloping surfaces that may be useful in achieving closer fitting assemblies. This algorithm could be of use in flexible manufacturing systems, automated assembly, automated selective

a s s e m b l y , e t c . , w h i c h a l so u s e c o o r d i n a t e m e a s u r e -

m e n t da t a .

N o a t t e m p t is m a d e in th is p a p e r to c o m p a r e the

p e r f o r m a n c e o f t he p r e s e n t a l g o r i t h m in t e r m s o f the

n u m b e r o f i t e r a t i o n s a n d the t i m e t a k e n , s i n c e a t rue

b e n c h m a r k is no t a v a i l a b l e fo r th is p u r p o s e . S i n c e

the m e a s u r e m e n t s w i t h a C M M are t a k e n at d i s c r e t e

s a m p l e p o i n t s , the s a m p l e p o i n t s m u s t b e j u d i -

c i o u s l y s e l e c t e d . O t h e r w i s e , r e l e v a n t i n f o r m a t i o n

m a y b e m i s s e d a n d m a y r e s u l t in a c e r t a i n d e g r e e o f

d i s a g r e e m e n t a m o n g the m e a s u r e m e n t r e su l t s . T h i s

s i t u a t i o n m a y a l so l e a d to d i f f e r e n c e s in r e su l t s

o b t a i n e d w i t h C M M a n d g a g i n g . F u r t h e r w o r k is

t a k i n g p l a c e to a r r i v e at t he a p p r o p r i a t e s a m p l i n g

s c h e m e .

References 1. Technical Drawings: Tolerancing of Form, Orientation, Location

and Runout--Generalities, Definitions, Symbols, Indications on Drawing, ISO 1101-1983(E), International Standards Organization, 1983. 2. Assessment of Departure from Roundness, BS 3730:1964, British

Standards Institute, 1964. 3. Measurement of Out-of-Roundness, ANSI B89.3.1-1972, Amer-

ican National Standards, 1972. 4. M.S. Shunmugam, "Comparison Between Linear and Normal

Deviations of Forms of Engineering Surfaces," Precision Engineer- ing, Vol. 9, No. 2, 1987, pp. 96-102. 5. M.S. Shunmugam, "On Assessment of Geometric Errors,"

International Journal of Production Research, Vol. 24, No. 2, 1986, pp. 413-25.

6. D.G. Chetwynd, "Applications of Linear Programming to Engi- neering Metrology," Proc. Inst. Mech Engrs (London), Vol. 199, No. B2, 1985, pp. 93-100. 7. D.J. Whitehouse, "A Best Fit Reference Line for Use in Partial

Arcs," J. Physics. E., Vol. 6, 1973, pp. 921-24. 8. T.S.R. Murthy and S.Z. Abdin, "Minimum Zone Evaluation of

Surfaces," International Journal of Machine Tool Design Research, Vol. 20, No. 2, 1980, pp. 123-36.

9. M. Fukuda and A. Shimokobhe, "Algorithms for Form Evalua- tion Methods of Minimum Zone and the Least Squares," Proc. Int. Sym. on Metrology and Quality Control in Production, Tokyo, 1984, pp. 197-202. 10. E.L. Stiefel, "Numerical Methods of Tchebycheff Approxima- tion," On Numerical Approximation, University of Wisconsin Press, Madison, 1959, pp. 217-32. 11. P.B. Dhanish, "An Algorithm for the Evaluation of Form Errors Based on the Theory of Discrete and Linear Chebyshev Approxima- tion," M. Tech Report, I.I.T Madras, India, 1988. 12. M.S. Shunmugam, "Unified Approach for Evaluation of Geo- metric Errors from Co-ordinate Measurements," Int. Syrup. on Mea- surement Technology and Intelligent Instruments, China, May 1989, pp. 98-105.

Authors' Biographies Dr. M.S. Shunmugam is Professor of Mechanical Engineering,

Indian Institute of Technology, Madras, India. Professor Shun- mugam received his PhD with metrology as his specialization. He has contributed over 80 technical articles and his publications span metrology, manufacturing processses, manufacturing systems, and computer applications in manufacturing. He is presently with Michigan Technological University, Houghton, MI as a visiting faculty member.

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establishing reference figures for form evaluation of engineering surfaces

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