ELSEVIER PIh S0263-2241 (96)00002-4

Measurement Vol. 17, No. 1, pp. 17 27, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in The Netherlands. All rights reserved 0263-2241/96 $15.00 +0.00

Intereomparison of parallelism measurements

R. Thalmann

Swiss Federal Office of Metrology, Lindenweg 50, CH-3084 Wabern

Abstract

An interlaboratory comparison of parallelism measurements was carried out among five national metrology institutes and four manufacturers of form measuring instruments from six European countries. The round robin lasted about 1 year. The comparison involved the measurement of the straightness of four generating lines and of the parallelism of opposite measurement profile pairs of a cylindrical steel standard and a rectangular ceramic straightness standard. The results show good agreement for the straightness measurements. The comparison of the parallelism measurements was not entirely satisfying, not only with respect to the stated measurement uncertainties, but also with respect to today's manufacturing tolerances. The analysis of the measurement results - - taking the measurement methods and instruments used into account - - documents the state of the art of cylinder form measurement and allows identification of problems and provision of recommendations regarding the measurement of parallelism. Copyright © 1996 Elsevier Science Ltd.

Keywords: Form measurement; Instrument errors; Calibration standard

I. Introduction

The measurement of parallelism is one of the key issues of precision form measurement, in partic- ular for the evaluation of cylinder form errors. Various commercial form testers offer this measure- ment capability, but a number of dedicated instru- ments were also developed for the measurement of straightness and parallelism of generating lines, especially at national metrology institutes.

The state of the art of roundness, straightness and parallelism measurements on an industrial level was reflected in the report of a previous intercomparison [ 1 ]. The agreement of the paral- lelism measurement results was considered to be rather unsatisfactory (the dispersion of the results was in the order of:___2 ~tm for a deviation from straightness of 2 lam), but the related possible prob- lems could not be identified.

The main objectives of this intercomparison were to document the state of the art of straightness and parallelism measurement, to improve paral- lelism measurement capabilities, to detect potential problems of measuring instruments, to investigate the evaluation of uncertainties and to harmonize the evaluation methods.

Three carefully selected standards of different form, size and material have been circulated between nine European industrial laboratories and national metrology institutes. The intercomparison was piloted by the Swiss Federal Office of Metrology. A list of participants and the corre- sponding measurement dates is given in Table 1. Following the detailed guidelines, the participating laboratories had to supply - - in addition to the required measurement results - - a description of the measuring instrument used as well as of the measurement method, and of the processing of the

18 R. Thalmann

Table 1 Participating laboratories and dates of measurement

Participants Date of measurement

Eidgen6ssisches Amt for Messwesen, OFMET, Wabern, Switzerland

08/93 12/93 06/94 09/93 09/93 10/93

Robert Bosch GmbH, Stuttgart, Germany FAG Kugelfischer, Schweinfurt, Germany Feinpriif Perthen GmbH, G6ttingen, Germany

Sveriges Provnings- och Forskningsinstitut, 11/'93 SP, Boras, Sweden

Physikalisch-Technische Bundesanstalt, PTB, 01/94 Braunschweig, Germany

Rank Taylor Hobson Ltd, Leicester, U.K. 02/94 subcontracted to NAMAS accredited calibration laboratory

Laboratoire National d'Essais, LNE, Paris, 03/94 France

lstituto di Metrologia "G. Colonnetti", 04/94-05/94 IMGC, Torino, Italy

measurement data. Importance was attached to the evaluation of error sources and the estimation of the measurement uncertainty.

2. International standards related to form measurement

The methods for the assessment and the meas- urement of deviation from roundness are described in the standards ISO 4291 and ISO 6318 [2,3]. Up to now, there was no corresponding standard available for straightness and parallelism measure- ments. However, for straightness the definitions and procedures can be treated in analogy to the roundness standards. Most manufacturers of com- mercial form measuring instruments have imple- mented this approach, which is also described in the test procedures of DIN [4]. Deviation from straightness can thus be evaluated with respect to the least squares (LS) regression line or with respect to the minimum zone (MZ) criterion (Fig. 1 ). While LS is restricted to computer evalua- tion, the MZ is as well suited for manual (graphi- cal) evaluation and better meets the functionality of the tolerance zone of ISO 1101 [5] .

J T

MZ

Fig. I. Straightness evaluation with respect to least squares (LS) and minimum zone (MZ) references.

The evaluation of parallelism by means of modern measuring instruments (where profiles are measured and stored digitally) is not sufficiently defined in existing international standards. Parallelism can be expressed in units of angle as a difference in direction or in units of length as a deviation of one measured profile from a straight line in the direction of a reference. The reference is defined by the second profile. For both profiles, LS and MZ evaluation can be applied. This yields three meaningful possibilities for the parallelism evaluation: LS/LS*, MZ/LS*, MZ/MZ*, where * denotes the reference, as sketched in Fig. 2. Whereas the latter two evaluations yield zones in length units, LS/LS* is expressed by an angle and is symmetrical, i.e. it gives the same result for both profiles to be taken as the reference. In addition, it is more stable than other evaluation methods, because it does not depend on peaks in the mea- sured profiles. For this reason, within this inter- comparison LS/LS evaluation was asked to be evaluated. The sign of LS/LS parallelism results is conventionally defined to be positive, when the measured profiles of the generating lines approach each other from the beginning to the end.

3. Description of the transfer standards

Three form standards were circulated. One of these, a cylindrical steel standard of 300 x 75 mm size, was new and turned out to be unstable during

R. Thalmann 19

MZ/LS*

MZ/MZ*

Fig. 2. Different ways of parallelism evaluation.

the intercomparison (bending of a few tenths of a micrometrej. Since no additional, useful informa- tion can be gained from the measurement results of the unstable cylinder, the discussion within this paper will be restricted to the two other transfer standards:

Figure 3 shows the cylindrical steel standard, 100 mm in length, 20 mm in diameter, with ground flats for stylus calibration.* The standard was provided with an appropriate support. The design of the standard is an outcome of a previous inter- comparison [1] , adapted to the usual size of workpieces from Rob. Bosch Company, where 25 such standards are routinely used for performance assessment and calibration of form measurement instruments. They are also used in other companies and serve as a calibration standard within the German calibration service DKD.

Figure4 shows the rectangular straight edge

* Manufactured by Rank Taylor Hobson Ltd, Leicester, U.K. and kindly loaned from Rob. Bosch Company, Stuttgart, Germany for the purpose of this intercomparison.

standard made of aluminium oxide ceramics, size 350 x 50 x 50 mmt. The two adjacent side faces A and B were manufactured with best quality (straightness <0.2 I, tm), the opposite faces are not specified and are typically 4 times worse with respect to form deviation and twice as bad regard- ing surface roughness.

Prior to the round robin, the transfer standards were carefully characterised regarding their suit- ability for such an intercomparison. Apart from surface roughness measurements, form measure- ments were made under different conditions (different filtering and different form error evalua- tion procedurest, and also on adjacent profile lines to check for homogeneity. The results of these acceptance test measurements helped to define the measurement conditions which were required from the participating laboratories. In addition, it allowed for a thorough measurement uncertainty estimation.

The stability of the standards was observed from the calibration results of the pilot laboratory before, in between and after the round robin. Table 2 shows the results of straightness measure- ments in August 1993, December 1993 and June 1994. The agreement is well within the measure- ment uncertainty, therefore no evidence for any instability of form was found. Note, that this was not the case for a third standard (300 mm steel cylinder, results not reported here), where a signifi- cant form change was observed. This third stan- dard was fabricated only a few months before the intercomparison. In addition, it had to be galvani- cally coated with a nickel layer in order to prevent corrosion. The one or the other fact probably induced the observed bending of the cylinder, which stabilized after approximately 18 months.

4. Measurement programme

The following measurements had to be carried out on the two standards: straightness measure- ment on four generating lines and parallelism evaluation on both pairs of opposite measurement

t Manufactured by TOTO Precision Ceramics. Kitakyushu- City, Japan.

20 R. Thalmann

150

= Measurement Length 100

t f Evaluation Length 90 l ~'~ 25 . . . . ~, ,

I IJ

Fig. 3. Cylindrical steel standard.

Evaluation length 320

TOTO \ - \ \

c \ Measurement line D

Fig. 4. Rectangular straightness standard of a luminium oxide ceramic.

Table 2

Stability measurements of the transfer standards

Standard Generator Straightness/~tm Uncertainty/ p.m

Date of calibration

08/93 12/93 06/94

Cylindrical 0 ° 0.06 0.06 0.07 0.04 s tandard 90 ° 0.07 0.05 0.08 0.04

180 ° 0.07 0.06 0.06 0.04 270 ° 0.06 0.06 0.07 0.04

Rectangular A 0.16 0.20 0.18 0.06 s tandard B 0.17 0.21 0.22 0.06

C 0.70 0.68 0.67 0.10 D 1.20 1.10 1.10 0.55

For the cylindrical standard, all the additional measurements, which are usually carried out in the calibration procedure at Rob. Bosch Company were proposed, i.e. roundness measurements on the cylinder at three different heights, roundness measurements at the two heights where the flick flats are located (see Fig. 3), and straightness meas- urements on the 45 ° generating line across the two flick flats. The results of all the optional measure- ments are not reported here. It has to be mentioned, however, that the differences in the parallelism results reported later in this paper, cannot be understood in view of the rather small calibration differences, seen in the flick results.

profiles. The measurement and evaluation lengths can be seen from Figs 3 and 4. Unfiltered LS evaluation was the minimum requirement. Additional filtering and MZ evaluations were optional.

5. Measuring methods and instruments of the participants

A short description of the measuring instruments and the measurement procedures of all participat-

R. 'Fhalmann 21

ing laboratories will be given. The laboratories are subsequently numbered Lab. 1 to Lab. 9 (Lab. 1 is the pilot laboratoryl . The first four laboratories (all of them are national metrology institutes) applied error separation techniques and used meas- uring instruments on which the specimen was supported horizontally. Laborator ies 5 -9 used commercial cylinder form measuring instruments, on which the specimen was measured in its vertical position on a rotat ing worktable. Table 3 gives an overview of the inst rumentat ion used. Lab. I. The long horizontal axis of a coordinate measur ing machine served as the straightness datum. The specimen was fixed horizontal ly between centres or clamped on the moving table. The other axes were used to posit ion two inductive probes which synchronously measure the profiles of two opposite generating lines. Rota t ion of the specimen a round its (cylinder) axis by 180 ° allowed for separation of the errors of the straightness axis and the component . Lab. 2. The long horizontal x-axis of a universal measuring machine (3 linear axes and one round- ness spindle axis, integrated in the vertical straight- ness axis) served as the straightness datum. The specimen was horizontal ly supported at the Bessel points on two V-blocks or flat blocks on the moving table. Parallelism was measured by succes-

sive straightness measurements on two opposite generating lines (the specimen is probed laterally) by turning the t ransducer at the spindle axis by 180 ° . The straightness measurements were cor- rected by a prestored da tum error, which was evaluated from repeated reversal measurements of a zerodur straightness standard. Lab. 3. Same universal measuring machine as Lab. 2. Specimen placed horizontally, fixed between centres or clamped to the table of the moving carriage. The specimen was probed laterally. For parallelism measurements, the second (opposite) generating line was probed after displacing the specimen using the y-axis. Error separation by a reversal technique was applied for the straightness, but not for the parallelism measurement, where a possible yaw error in the y-axis was not under control. Lab. 4. Horizontal straightness measurement bench with ceramic air bearing straight guide. The cylin- drical s tandard was supported on V-blocks, the rectangular one was kinematically mounted on three balls. The specimen was in a fixed position while being probed laterally by a moving trans- ducer. Error separation using a reversal technique was applied for straightness, but not for parallelism measurement. Lab. 5 and Lab. 6. Form measurement instrument

Table 3 Measurement instruments of the participating laboratories

Laboratory Type of instrument Instrument axes Error separation

Lab. 1 Modified CMM 3 linear axes Yes (reversal technique) Long horizontal axis used for straightness datum; other axes for positioning the probes 3 linear and 1 roundness axes. x-axis used for - - and II measurement

Lab. 2 Modified universal measuring machine

Lab. 3 Modified universal measuring machine

Lab. 4 Straightness measurement bench

Lab. 5 Commercial form measurement instrument

Lab. 6 Same as Lab. 5 Labs 7 9 Similar to Lab. 5

3 linear and 1 roundness axes. x-axis used for - - and II measurement 1 horizontal axis

3 axes (1 vert., 1 hor., 1 rot.). Vert. axis used for and II measurement

Same as Lab. 5 Same as Lab. 5

Yes, but indirectly Icorrection data file obtained from reversal measurement of zerodur straightness standard) Yes

Yes (for straightnesst

Yes, but indirectly (correction data file obtained from reversal measurement of cyl. steel standard) Same as Lab. 5 No

22 R. Thalmann

with air bearing rotating work table and one horizontal (radial) and one vertical (straightness datum) axis. The standards were measured in their vertical position, kinematically mounted. The straightness error of the vertical axis was software corrected. The correction was evaluated from reversal measurement of a steel cylinder. The so called "true parallelism" method was applied for the parallelism measurement, i.e. the specimen remained fixed while the probe successively mea- sured two opposite generating lines (the radial axis permitted to displace the stylus above the specimen through the centre in order to probe from the opposite side). In order to eliminate residual sys- tematic errors in parallelism, the average was taken from two measurements with the cylinder in a normal and in an upside down position. Lab. 7. Similar instrument to Lab. 5. No error separation was applied. "True parallelism" measurement. Lab. 8. Similar instrument to Lab. 5. No error separation was applied. Parallelism was measured by two successive straightness measurements with rotation of the specimen by 180 ° in between. Lab. 9. Similar instrument to Lab. 5. For the cylindrical standard, no error separation was applied. The parallelism was measured by two successive straightness measurements with rotation of the specimen by 180 ° in between. For the rectan- gular standard, the straightness and parallelism were measured simultaneously using two probes. Error separation was applied using the reversal method after rotation of the specimen by 180 ° .

6. Measurement results and discussion

The comparison and discussion of the measure- ment results will be mainly based on their graphical representation. In the figures, the results are shown in the order of the laboratory numbers used in the preceding section. From the pilot laboratory, only the results of the intermediate calibration, which is regarded to be representative, were introduced in the figures. The other results of the pilot labora- tory can be taken from Table 2, where the stability of the standards was demonstrated. If a laboratory did not contribute to the results, the appropriate space is blank in the figure. In Fig. 5 the profile

lines of a typical measurement of the pilot labora- tory are represented. It shows the straightness and parallelism deviation of two generating lines of the cylindrical standard and all four measurement lines of the rectangular standard.

For the following reasons, no reference value was determined to compare with the measurement results. The pilot laboratory did not claim to give the values nearest to the true values and the uncertainty estimates were evaluated too differently for taking the result with the smallest claimed measurement uncertainty as a reference value. Furthermore, the average over all results does not necessarily represent the best knowledge of the measurand due to the large dispersion of some results and the large differences of the claimed uncertainties. Therefore, only the spread (maxi- mum-minimum value) and the standard deviation will be considered for quantitative comparison.

6.1. Straightness measurements

Figures 6 and 7 show the straightness measure- ment values together with the uncertainty bars of the two standards for the unfiltered LS evaluation. From Lab. 4, no unfiltered values were available, therefore filtered results for 0.8 mm and 2.5 mm cut-offs were taken. The measurements of the cylin- der standard show good agreement. A trend to somewhat larger straightness values can be explained either by noise (Lab. 3) or by the influ- ence of the larger form error in the zone close to the edges which had obviously not been completely excluded for the evaluation. For the straight edge, the results for the face D (which is not specified by the manufacturer and has no functional use) agree significantly less, but improve with filtering, as can be seen from the last group of measurements obtained with an 8 mm cut-off filter. From an analysis of the measured profiles of the line A which showed very small straightness deviations, the (residual) guide errors within the measurement range used for this comparison were estimated to be smaller than 0.05/am for those who applied error separation and between 0.05 and 0.4/am for the other laboratories.

Table 4 shows the spread and the standard devia- tion of all straightness measurement results for the two standards. The agreement seems to be depen-

R. Thalmann 23

Cylindrical Standard 0°/180 °, LS/LS +0.3 pm

0 ~ . . . . . . . . . . . . . . . - . . . . . . . ~ . . . . / Straightness deviation 0.06 pm

-0.3 pm

Straightness deviation 0.07 IJm

(a)

• /

Parallelism 0.24 pm/m

\ l +0.3 pm

0

-0.3 IJm

(b)

Rectangular Standard A/C, LS/LS ,,

Straightness deviation 0.15 pm / i

Parallelism 0.44 pm/m ', i

i . I ~ Straightness deviation 0.67 pm

fRectangular Standard BID, LS/LS +0.5 pm i

0 - - - ~ '" ' ' / -- //__- ' - ' - ~ - ~ , , - [. Straightness deviation 0.16 pm / i -0.5 pm

~ ~ ~ , J ~ l ,I ' ParaUelism'6.21t'lm/m i

Straightness d/eviatio n 1.31 pm ~ I~ql ~ 1 1 ~

(c)

Fig. 5. Measurement profiles of two generating lines of the cylindrical standard (a) and of all four measurement lines of the rectangular standard (b,c). The dashed lines indicate the evaluation zone.

dent on the actual form of the profile rather than on the measurement length.

6.2. Parallelism measurements

In Figs 8 and 9, the parallelism measurements of the two standards are shown for unfiltered LS

evaluation. From Lab. 4, again filtered values were taken. The differences between the maximum and the minimum values, and the standard deviations are represented in Table 5. As can be expected, these values are approximately inversely propor- tional to the measurement length. The standard deviation is about 0.5 pm/L, where L is the meas-

24 R. Thalmann

0.9 E

0.8

~, 0.7

~0.6 ® "~ 0.5 ~OA ~0.3

~ 0.2 (h 0.1

0

0 o 9 0 ° 180 ° 2 7 0 °

Fig. 6. Straightness measurements of the cylindrical standard for the four generating lines, unfiltered LS evaluation. (Within each group of measurements, the order on the abscissa corresponds to participant numbers 1 9.)

3

~ 2.5

2 .$:

'~ 1 .5 .

D.

; t ;T

D, 8 mrn cutof f

i" it- Fig. 7. Straightness measurements of the rectangular standard

Table 4 Agreement of straightness comparison

Standard Generator (Max-Min)/ Standard deviation/ ~m lam

Cylindrical 0 ° 0.24 0.07 standard 90 ° 0.25 0.07

180 ° 0.19 0.07 270 ° 0.18 0.06

Rectangular A 0.23 0.08 standard B 0.44 0.13

C 0.56 0.20 D (1.5) (0.52)

for the four measurement faces, unfiltered LS evaluation.

u rement length. It can be seen from the figures, that for most of the labs, the observed differences in the measur ing results seem to be s imilar for each pa i r of genera t ing lines for the same s tandard . The differences thus or iginate from a systematic ins tabi l i ty or an e r ror in the geomet ry of the measur ing inst rument . Also an ins tabi l i ty of the moun t ing device of the cyl inder s t anda rd canno t be excluded.

In addi t ion , the fol lowing pa r t i cu la r observa- t ions can be made: Labs. 1, 2 and 4 ob ta ined very s imilar results, most ly in agreement within their measuremen t uncer ta in ty of 0.2-0.6 ~tm/m. Lab. 3 ob ta ined sys temat ica l ly smal ler values by abou t 3 p.m/m. The cause might be a yaw er ror in the y- axis while d isplac ing the s t anda rd between the

R. Thalmann 25

11 180~fO °, LS/LS

'°t 9 -

E 7

2 i T 32

o I T -

-2 ~ • -3 ~ ! -4 !

T i L

270~'90 o, LS/LS

t i

± ~-

I L

4 i I

I i

Fig. 8. Parallelism measurements of the cylindrical standard, LS evaluation, unfiltered. (The arrows indicate measurement results lying outside the represented area.}

8

°t 4

E 2

~ -2 -4

-6 ;

~" -8 i -10

-12

A/C, LS/LS

, -

I t

B/D, LS/LS

" I "" • I . , . _ - "" I

i

Fig. 9. Parallelism measurements of the rectangular standard. LS evaluation, unfiltered.

Table 5 Agreement of parallelism comparison

Standard Generator (Max-Min)/ Standard deviation/ ~Lmm I r tmm-1

both within 1 2 pm/m from Labs 1 and 2. Labs 7, 8 and 9, who used similar instruments to Labs. 5 and 6, but without applying the reversal technique, got significantly larger deviations.

Cylindrical 180:/0 19.8 4.7 standard 270':/90' 24.2 5.7

Rectangular C/A 9.5 2.5 standard D/B 5.7 1.5

measurements of the two generating lines. Labs 5 and 6, who used the same instrument and the same measurement technique (measuring the specimen in two positions, normal and upside down, and taking the average) got very similar results and are

7. Measurement uncertainty

The participants were asked to report all meas- urement results together with their associated uncertainty. The statement of uncertainty should be consistent with the ISO-Guide [6] . This requires the uncertainty to be given as an estimated combined standard uncertainty u obtained by com- bining the individual standard uncertainties result-

26 R. Thalmann

ing from type A and type B evaluation. For straightness measurements, the following error contributions were taken into account: • Reproducibility • Probe calibration • Probe linearity • Surface roughness • Guide straightness deviation • Errors in the measurement and evaluation

length • Filter sampling.

A sound uncertainty estimation for parallelism appeared to be difficult. It depends, of course, to a certain extent on the straightness measurement uncertainty. All other (and in general more important) contributions have to be estimated on the basis of repeatability and reproducibility measurements, instrument stability and drift considerations.

Evidence for the difficulties of uncertainty esti- mation in form measurement is obtained from the figures illustrating the measurement results together with error bars. On the one hand, too many measurements obviously do not contain the estimated true value within their expanded uncer- tainty interval, on the other hand some laboratories seem to have overestimated the measurement uncertainty. The results of this intercomparison will certainly help the participants to improve their measurement uncertainty estimations.

8. Conclusions

The observations of a previous intercomparison [ 1 ] have been confirmed: satisfactory accordance can be obtained from measurements of single meas- urement lines (straightness), whereas large varia- tions are found, when position and form deviation have to be combined (parallelism). Using modern form measuring machines with well corrected datum axes, a straightness measurement accuracy of 0.1-0.2 I-tm is usually achieved on appropriate standards. With the help of error separation tech- niques, an accuracy even better than 0.05 ~tm can be obtained, almost independent of the measure- ment length. Regarding the parallelism measure- ment, the agreement within this intercomparison

is up to 10 times worse, if expressed as the 2-fold standard deviation of the dispersion of the meas- urements, and amounts to approximately 1 ~tm/L, where L is the measurement length.

Several instruments suffer from systematic errors originating from instabilities, geometry errors or insufficient calibration and correction of the machine geometry. It has been shown, that in general, dedicated instruments for straightness and parallelism measurement show better performance than multi-purpose form testers. It would however be unfair to try to identify certain measuring instruments to be used preferentially for parallelism measurement, since we cannot compare expensive universal measuring machines, specialized instru- ments dedicated for parallelism measurement, and commercial form testers. But certainly the conclu- sion holds, that the so-called true parallelism meth- ods operating without moving the workpiece in between the two measurements of the generating lines, are to be preferred.

The two standards, which have both been manu- factured to form deviations smaller than the meas- urement uncertainties of most of the participants, were proven to be perfectly well suited as transfer standards for both straightness and parallelism. The manufacturers" capability for such small form tolerances underlines the need for further improve- ment of form measurement instrumentation. In particular the cylindrical standard turned out to be useful for checking the performance of form measuring instruments. Within half an hour it is possible to assess the roundness, straightness and parallelism deviations of the instrument, to cali- brate the transducer and to check the filtering. The mounting device for the cylinder should, however, be further improved. It might have caused certain instabilities in the parallelism measurements.

The overall benefit of this comparison can be summarized with the following recommendations: • Investigation of the stability of form measure-

ment instruments, in particular between the rotational axis and the straightness column.

• Improvement of calibration and software cor- rection procedures of the instrument/machine geometry.

• The regular use of transfer standards to check the measurement performance of the instrument.

R. Thahnann 27

• To check for systematic parallelism errors by inverting the position of the standard.

• To include the definition and measurement of parallelism in future written standards.

• Uncertainty estimations should be done in accordance with the ISO-Guide for the Expression of Uncertainty in Measurements [6].

Acknowledgements

The author wishes to thank all his colleagues for their fruitful contributions and comments and Mr Jt~rg Spiller from OFMET for carrying out the measurements. The intercomparison has been financially supported by the Commission of the European Communities under its Programme for Applied Metrology and Chemical Analysis (Community Bureau of Reference, BCR) [7].

References

[ I ] H. Bosse, F. Lfidicke and H. Reimann, An mtercomparison on roundness and form lneasuremenl, ,'~[ea,surt,ment 13 (1994) 107.

[2] ISO 4291 1985, Methods for the assessment of departure from roundness: Measurement of variations in radius, International Organisation of Standardisation, Geneva, Switzerland, 1985.

[3] ISO 6318-1985, Measurement of roundness: Terms, defini- tions and parameters of roundness, International Organisation of Standardisation. Geneva. Switzerland, 1985.

[4] W. Aberle. B. Brinkmann and tt. MOiler, Prz~li'erlahren /iir Form- und Lageahweichun'zen, Beulh Kommentare, 1983.

[5] ISO 1101-1983, Technical drawings, geometrical toleranc- ing: Tolerances of form, orientation, location and runout: Generalities, definitions, symbols on drawings, International Organisation of Standardisatiog Geneva, Switzerland. 1983.

[6] Guide to the Expression ol Uncertainty in Measurement, 1st edition 1993, corrected and reprinted 1995, International Organisation of Standardisation, Geneva, Switzerland.

[7] R. Thalmann, Intercomparison of parallelism measure- ments. BCR lnlbrmation Applied Metrolo~v, Report EUR 16162 EN 1995.