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International Journal of Machine Tools & Manufacture 47 (2007) 1229–1236

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Evaluation of form data using computational geometrictechniques—Part I: Circularity error

N. Venkaiah, M.S. Shunmugam�

Manufacturing Engineering Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai—600036, India

Received 12 March 2006; received in revised form 12 August 2006; accepted 15 August 2006

Available online 2 October 2006

Abstract

The present work deals with evaluation of form error from the measured profiles obtained using a form tester, namely roundness/

cylindricity measuring instrument. In Part I, details of circularity evaluation are presented. Due to eccentricity in component setting and

radius-suppression inherent in the measurement, circularity error has to be evaluated with reference to a limacon. A computational

geometry-based algorithm is proposed for establishing minimum circumscribed, maximum inscribed and minimum zone limacons. A new

type of control hull for directly constructing equi-angular diagrams and a new procedure for updating are introduced. Validation has

been done with bench-mark data set and corresponding results available in the literature. Being geometry-based algorithm, it is simple to

follow and each iteration can be visualized and interpreted geometrically. On comparison with simplex search method, the proposed

algorithm is found to be computationally efficient in terms of accuracy and time taken. The proposed methods can be easily implemented

in computer-aided roundness measuring instruments. Extension of this work for evaluation of cylindricity error has been dealt in Part II.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Circularity error; Form data; Limacon; Computational geometry; Control hull; Equi-angular diagrams

1. Introduction

The geometric form of any manufactured feature alwaysdeviates from its nominal design to some degree, owing tothe random and/or systematic errors. In order to satisfycertain functional requirements or assembly conditions,geometric tolerances are usually assigned to selectedfeatures. One such feature often used in engineeringcomponents is a cylindrical feature. In certain applications,it is enough to consider a transverse section of the cylinderand apply circularity tolerance. Verification of thistolerance is done using a roundness tester, which measuresthe deviations from an ideal circular trajectory establishedby the instrument. In case of cylindricity measurement, anadditional straight datum is used and the measurements arecarried out at few transverse sections of the cylinder. Insuch measurements, the size/radius of the component isalways suppressed and the measurement data is referred to

e front matter r 2006 Elsevier Ltd. All rights reserved.

achtools.2006.08.010

ing author. Tel.: +9144 22574677; fax: +91 44 22570509.

ess: shun@iitm.ac.in (M.S. Shunmugam).

as form data. The form data is usually characterized byequi-spacing.The profile deviations obtained using roundness measur-

ing instrument are magnified and plotted on a polar chart.The earlier manufacturers of the instruments providedtemplates with concentric circles to evaluate the circularityerror. Whitehouse [1] and Chetwynd [2], however, showedthat due to eccentricity between the rotational center of theinstrument and center of the component, even a trulycircular component results in a non-circular profile.Evaluation using a pair of concentric circles would showcertain circularity error and an error value greater than thespecified tolerance might lead to the rejection of thecomponent. An exact mathematical representation of themeasured profile obtained for a truly circular componentfor a given eccentricity, radius suppression and magnifica-tion was obtained and the first-order approximation wasshown to be a limacon [3]. Therefore, limacon would be theappropriate assessment feature for evaluation of circularityerror. Geometrically, a limacon is obtained as a locus ofthe foot of the perpendiculars drawn from the origin (pole)to the tangents of a circle having its center offset from the

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Nomenclature

ei deviation of ith point from assessment limaconi index for data pointsri, yi polar coordinates of ith pointro radius of circle for assessment limaconxo, yo center coordinates of circle for assessment

limaconCf farthest centerCn nearest centerEA equiangular line

FE farthest edgeLO edge of an outer control hullLI edge of an inner control hullLS least squaresMC minimum circumscribingMI maximum inscribingMZ Minimum zoneNE nearest edgeV vertex on a control hullD form (circularity) error

N. Venkaiah, M.S. Shunmugam / International Journal of Machine Tools & Manufacture 47 (2007) 1229–12361230

origin. Though the templates with concentric circles are nolonger provided with roundness testers, the terms such asleast squares circle (LSC), maximum inscribed circle(MIC), minimum circumscribed circle (MCC) and mini-mum zone circle (MZC) are quite liberally used by theindustry while specifying the method used for evaluatingthe circularity error.

ISO specifies that an ideal feature must be establishedfrom the measured profile such that deviation between itand the measured profile is the least possible value [4]. Inpractice, the ideal feature is taken to be a straight line forstraightness, a plane for flatness, etc. Some researcherssimply extend this to circularity evaluation without anyconsideration to the method of measurement and use acircle as an assessment feature for the form data. Even theuse of the circle is justified with a claim that the limacon isan approximation of a circle and therefore only the circlemust be used for obtaining correct results. In some cases,even though the center coordinates of circle for establishingthe limacon are obtained using least squares method(LSM), two concentric circles are drawn from this centerenclosing the entire measured profile and circularity error isevaluated as radial separation between the two circles. Thefact that the limacon is the basis for the LSM and theparameters evaluated actually correspond to the limacon isoften ignored [5].

A number of attempts have been made in the past toevaluate circularity error from the form data obtainedusing roundness measuring instruments with limacon as theassessment feature. LSM is one such attempt based on asound mathematical principle that minimizes the sum ofthe squared deviations of the measured points from thefitted limacon feature [6,7]. This method is robust, but itdoes not follow the standards intently and will notguarantee the minimum zone (MZ) solution specified inthe standards. The deviation values and the form error thatare determined by LSM will be generally larger than theactual ones, and this may lead to rejection of good parts.To obtain the MZ solution in circularity evaluation, thenumerical methods based on simplex linear programming[8] and simplex search [9] have been adopted. Shunmugam[6] suggested a new simple approach called the mediantechnique, which gives minimum value of circularity error.

Using discrete Chebyshev approximations, Danish andShunmugam [10] have arrived at the MZ values.In the past decade, computational geometric techniques

have gained enormous attention from the designers ofalgorithms for solving geometric problems [11,12]. Thesetechniques also show greater promise for solving the MZproblems encountered in the geometrical evaluations. Theycan be very well applied for the evaluation of the circularityerror in the manufactured components. Though applicationof Voronoi diagrams for construction of circle has beenreported in the literature, first attempt to develop computa-tional geometric techniques based on limacon was reportedby Samuel and Shunmugam [13]. Two limacons obtainedfrom the same center, enclosing the entire measured profileand having minimum separation, result in an MZ value. Theconcepts of function-oriented evaluation, namely methodsbased on minimum circumscribed (MC) and maximuminscribed (MI) limacons, are also applied for circularityevaluation. In order to obtain MC and MI limacons, convexouter and inner hulls have been used in the past. Constructionof convex hulls considerably reduces the number of candidatepoints for establishing the assessment limacons. However,there is a need to reduce the candidate points further in caseswhere a large number of data points are to be evaluated. It isalso observed that the algorithms used to construct theconvex inner hull are not quite consistent [5,13].Part I deals with the evaluation of circularity error

bringing out a new concept of control hull which leads toconsistent inner hulls and reduces the number of candidatepoints for establishing the MC, MI and MZ limacons. Anew procedure for updating is also introduced. The resultsobtained for the circularity data reported in the literatureare included in this part. In comparison with simplexsearch method, the proposed algorithm takes lesser time,leads to higher time saving for large datasets, and givesaccurate results in absence of any convergence criterion.Part II of this paper deals with the cylindricity evaluationfrom the form data [14].

2. Circularity error

Fig. 1 shows a point on the roundness profile representedby {ri, yi}. The figure also shows a limacon used for the

ARTICLE IN PRESSN. Venkaiah, M.S. Shunmugam / International Journal of Machine Tools & Manufacture 47 (2007) 1229–1236 1231

assessment, as a curve representing the locus of foot of theperpendiculars drawn from origin O (pole) to the tangentsof a circle whose radius is ro and center is Oo(xo, yo) [13].Fig. 2 shows the deviation of a given point Pi from theassessment limacon. For quantifying this deviation ei,another limacon passing through Pi may be assumed andthe circle from which this limacon is obtained has the samecenter, but different radius. It can be clearly seen that theseparation between these two limacons is a measure of thedeviation of point Pi from the assessment limacon.Interestingly, the radial distance between the two circlesfrom which the limacons have been generated is also equalto the deviation ei, but the point Pi falls outside thecorresponding circle. By convention, deviation of a

Fig. 1. Circularity data and limacon (for illustration only).

Fig. 2. Deviation of a p

measured point lying outside the assessment limacon istaken to be positive and a point inside is considered to havenegative deviation. The circularity error (D) is, therefore,obtained as absolute sum of the maximum and minimumdeviations.

3. Control hull

Crest and valley limacons pass through the extrememeasured points. Therefore, it was considered appropriateto construct corresponding convex hulls. Outer hull wasconstructed as smallest convex hull enclosing all themeasured points and inner hull was taken as largest emptyconvex hull. For each line connecting a vertex of theconvex hull and the origin, a perpendicular line was drawnand equi-angular lines were constructed at the intersectionof these perpendicular lines. It is observed that thealgorithms [5,13] found in the literature for the convexinner hull are not consistent in constructing the same.While analyzing several alternatives for constructingunique inner convex hull, it is realized that the perpendi-cular lines drawn at each measured point yield a polygon,which can be directly used for constructing equi-angulardiagrams. This polygon also results in the reduction of thenumber of candidate points for the construction of equi-angular diagrams and hence the subsequent computationaleffort.To differentiate between this polygon and the conven-

tional convex hull used in computational geometryliterature, the polygon obtained in the present work isreferred to as control hull. The new concept of forming acontrol hull followed in the present work is explained usingFigs. 3(a) and 4(a). As shown in Fig. 3(a), at each point onthe measured profile, a line is drawn perpendicular to theradial line joining the measured point and the origin,namely the instrument center. A smallest polygon is formed

oint from limacon.

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Fig. 3. Construction of farthest EA diagram: (a) outer control hull, (b) initial farthest EA edges, (c) updated hull and farthest EA edges, (d) farthest EA

diagram and MC limacon.

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by the intersection of these lines such that all the measuredpoints are confined within it. Such a polygon is referred toas outer control hull. Similarly, largest empty polygonobtained by the intersection of the perpendicular lines asshown in Fig. 4(a) is referred to as inner control hull.

Interestingly, outer and inner control hulls are also convexin nature.

4. MC limacon

Fig. 1 shows hypothetical profile of a circular feature.Outer control hull of the measured points is constructed asexplained earlier. The edges of the hull such as LO1, LO2,LO3, LO4 and LO5 are established by the measured pointsP1, P3, P6, P8 and P11, respectively, as shown in Fig. 3(a).

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Fig. 4. Construction of nearest EA diagram: (a) inner control hull, (b) initial nearest EA edges, (c) updated hull and nearest EA edges, (d) nearest EA

diagram and MI limacon.

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In the next step, for each pair of adjacent edges, equi-angular (EA) lines, such as EA12, EA23 etc are constructedat V12, V23, etc. as shown in Fig. 3(b). If a line EA23 isconsidered, it intersects with EA12 and EA34 at Cn (C12)and Cf (C34), respectively. The intersection point Cf, whichis the farthest from the vertex V23, is the farthest center(C23). A circle drawn with this farthest center and withsuitable radius will be tangential to the respective edges ofthe hull. For example, a circle with center Cf (C23) can bedrawn tangential to the lines LO2 and LO3. It should beremembered that the circle being referred to here representsthe circle from which the limacon is established. Thelimacon constructed from this circle would pass throughpoints P3 and P6. The portion of the line EA23 beyond Cf,away from V23 is the farthest EA edge (FE23) correspond-ing to the edges LO2 and LO3 of the hull. Following theprocedure outlined here, the farthest EA edges correspond-ing to all pairs of edges of the hull are constructed as shownin Fig. 3(b). It is seen in Fig. 3(b) that a few edges havecommon farthest centers. For example, farthest edges FE23

and FE34 have C23 and C34 as their centers, and for anypoint in the region enclosed by these two edges, the edgeLO3 is the farthest. The outer hull is therefore updated bydropping the edge LO3. In case of the conventional convexhull, a vertex is dropped while updating [5,13]. The edgeLO5 is also dropped and a new hull is formed as shown inFig. 3(c). New EA edges are formed for this updated hullfollowing the same procedure. The new edges FE12, FE24

and FE41 pass through a common center and therefore theprocedure comes to end at this stage. The completediagram as shown in Fig. 3(d) is referred to as farthestEA diagram. By taking the centers of the complete EAdiagram, a number of circles can be drawn for the givendata set. Out of these circles, one having the least radius ischosen to construct the MC limacon. The smallest circlethus obtained is tangential to the edges LO1, LO2 and LO4

of the hull with center at C124. For the sake of under-standing, the limacon established from this circle is alsoshown in Fig. 3(d), and this limacon passes through P1, P3,and P8.

ARTICLE IN PRESSN. Venkaiah, M.S. Shunmugam / International Journal of Machine Tools & Manufacture 47 (2007) 1229–12361234

5. MI limacon

Fig. 4(a) shows the inner control hull of the measuredpoints, constructed as explained earlier. This hull is used toconstruct the nearest EA diagram. The procedure forconstructing the nearest EA diagram is similar to that ofthe farthest EA diagram, except that the nearest intersec-tion points are considered as end points of the nearest EAedges instead of farthest intersection points. Fig. 4(b)shows the nearest EA edges for the initial hull. The nearestEA edge between LI2 and LI3 is obtained by consideringthree EA lines namely, EA12, EA23 and EA34. The nearestintersection point is taken as the nearest center Cn (C23),and the EA line toward the vertex V23 from this center isthe nearest EA edge NE23. The hull is updated by droppingthe nearest edges on the hull from the common nearestcenters. For example, the EA centers C23 and C34 arecommon, and hence the edge LI3 is dropped. Similarly, theedge LI1 is dropped, and a new hull is formed as shown inFig. 4(c). The complete nearest EA diagram and the MIcircle and corresponding limacon are also shown in Fig.4(d). It can be noted that the MI limacon thus constructedpasses through P5, P10 and P12 with its center at C245.

6. MZ limacons

Two concentric limacons with least possible separationbetween them and enclosing all the measured points aresaid to be MZ limacons. The MZ limacons are controlledby at least four points of the dataset. This can occur whenthree points lie on one limacon and the fourth point on theother limacon concentric to it or when two points are oneach limacon. The farthest and nearest EA (shown as

Fig. 5. Superimposed farthest and nearest EA diagrams with MZ

limacon.

dashed lines) diagrams are superimposed, as shown in Fig.5. The intersection points of these two diagrams are thecandidate centers [5]. The smallest possible circumscribingcircle and largest possible inscribing circle with center atthe intersection points of farthest and nearest EA diagramsare obtained and corresponding limacons are established.Such limacons will contain all the measured points betweenthem. In case an intersection point coincides with acommon center of either the farthest or nearest EA edges,one of the limacons would be controlled by three points.Otherwise, both the limacons are controlled by two pointseach. The radial distance between these concentric lima-cons is found, and a pair having minimum radialseparation gives the MZ limacons. The radial distancebetween these limacons is the circularity error. It isinteresting to note that the difference between the radiusvalues (ro) of the outer and inner circles corresponding tothe respective limacons also denotes the MZ error. Thecenter Oo obtained by the intersection of the farthest andnearest EA diagrams as shown in Fig. 5 represents thecenter corresponding to the circles for MZ limacons. Itmay be noted in Fig. 5 that the outer limacon passesthrough two points P3 and P8, whereas the inner limaconpasses through P5 and P10.

7. Results and discussion

The proposed algorithms were applied on different sizesof data sets. Inclusion of large sets in this paper requiresmore space and even in such cases, only a few significantpoints finally influence the control hulls. Also, the pointscontrolling the hull are not equi-spaced even for equi-spaceddata set. Therefore, it is considered prudent to include in thispaper a bench-mark data set taken from the literatureinstead of a large and new data set and to validate theproposed algorithms with the results published for thebench-mark data set. The bench-mark data set [6] shown inTable 1 is used in the present work and salient features ofthe proposed algorithm are brought out.Table 2 shows the results obtained on the basis of MC,MI and MZ methods and the corresponding roundnesserror (D) values. The outer control hull is formed by fourpoints only and it needed one update to get the completefarthest EA diagram. The MC circle is tangential to LO1,LO3 and LO4 and the limacon obtained from this circlepasses through the points 1, 4 and 6. The inner control hullis formed by six points necessitating two updates to get thecomplete nearest EA diagram. The MI circle is tangential toLI3, LI4 and LI5 and the limacon obtained from this circle

Table 1

Circularity data [6]

Point i 1 2 3 4 5 6 7 8

yi (deg) 0 45 90 135 180 225 270 315

ri (mm) 4 4 3 5 2 3 1 2

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Table 2

Results of circularity evaluation

Iteration Limacon

MC edge (point i) MI edge (point i) MZ edge (point i) LSM

Initial LO1(1), LO2(2), LI1(1), LI2(2), LI3(3), — —

LO3(4), LO4(6) LI4(5), LI5(7), LI6(8)

1 LO1, LO3 LI2, LI3, LI5, LI6 — —

2 — LI3, LI5, LI6 — —

Final LO1, LO3, LO4 LI3, LI4, LI5 LO1, LI3, LO3, LI4 —

(1, 4, 6) (3, 5, 7) (1, 3, 4, 5)

Parameters

xo (mm) 0.0000 0.0000 �0.1213 0.1464

yo (mm) 1.4142 1.0000 1.1213 1.2071

ro (mm) 4.0 2.0 3.0 (1.8787, 4.1213) 3.0000

D (mm) 2.4142 2.2929 2.2426 2.4571

N. Venkaiah, M.S. Shunmugam / International Journal of Machine Tools & Manufacture 47 (2007) 1229–1236 1235

passes through the points 3, 5 and 7. The MZ circles areobtained by the intersection of farthest and nearest EAdiagrams. The outer circle (max ro ¼ ro+D/2 ¼ 4.1213mm)is tangential to LO1 and LO3 and the limacon obtainedfrom this circle passes through the points 1 and 4. Theinner circle (min ro ¼ ro—D/2 ¼ 1.8787mm) is tangential toLI3 and LI4 and the limacon obtained from this circlepasses through the points 3 and 5. The parameters ofthe assessment limacons and corresponding error valueshave been included in the Table 2.

For the purpose of comparison, the values based onthe LS method are also included in Table 2. The equationsfor estimating LS parameters are given in Appendix A.The LS method yields higher values of circularity error.The evaluation of the circularity error by the computa-tional geometric techniques yields accurate results.The results obtained with the present method are inagreement with those reported results [13]. However,search methods reported in the literature for the evaluationof circularity error based on MZ approach requiresome value for convergence, as search is made in parameterspace, namely {xo,yo,ro}. The algorithms proposed inthe present work do not require any specified convergencevalue and the results depend only on the points in thedata set that are used for establishing the assessmentlimacons.

The values of the center position and size obtained by LSmethod show considerable statistical stability, because allthe measured points are included in the computation ofthese values. In case of MZ and function-orientedevaluation (MC and MI), the center position and size aredetermined by the extreme points. In the first instance, thefeatures established by the extreme points may seem tohave little practical significance. It should be rememberedhere that in any method used for establishing theassessment/reference features, the value of the circularityerror is finally computed on the basis of extreme pointsonly. Therefore, proper care must be exercised in all caseswith regard to the unstable extreme points and uncertaintyin their measurements.

In the proposed algorithms, the outer control hull andthe inner control hull of the measured points are taken sothat most of the points that do not influence the finalresults are eliminated, and the construction of EAdiagrams becomes less cumbersome. It is also observedthat the number of points on the outer/inner control hull isconsiderably lesser than that on its conventional counter-part, namely outer/inner convex hull, thereby minimizingthe computational effort.The computer programs for the proposed algorithms

were written in C++ and run on a Pentium (R) IV, 256MB RAM, 2.4GHz machine. For the bench-mark data set,LS method took 0.002 s, while simplex search method took112 iterations and 0.062 s using a convergence criterion of10�9 on standard deviation to get the final result of MZ. Theproposed method took 0.032 s for MZ evaluation of thesame data set, whereas MC and MI evaluations took 0.015 sand 0.016 s, respectively. These algorithms were tested forvarious data in the literature [6,13] and considerable savingin time is observed for large data sets. MC, MI and MZevaluations of 360-points data set by the proposed methodstook 0.128, 0.144 and 0.286 s, respectively.

8. Conclusions

The form data from roundness measuring instrumentshas to be treated differently from the data obtained fromcoordinate measuring machines. There is a radius-suppres-sion in the form data and a profile-distortion is introduceddue to eccentricity in setting the component on the table ofroundness measuring instrument. Hence, a limacon has tobe used as an assessment feature. In this work, computa-tional geometric techniques have been developed to fit MC,MI and MZ limacons. These techniques are simple andelegant, because the geometrical aspects can be visualizedmuch better than the mathematical aspects of thenumerical techniques. Since the details are explained witha simple data set, researchers can easily implement and testtheir algorithms.

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For fitting the limacons, EA diagrams are established onthe basis of a new concept of control hulls which aresimpler to construct and the number of points to be dealtwith are reduced considerably. Also, control hulls can bedirectly used to construct the EA diagrams and, as a result,computational effort is reduced. Following this, a newupdating procedure involving edges of control hull hasbeen introduced in this paper. The methods proposed forthe circularity error using MC, MI and MZ approacheshave been tested successfully on the data available in theliterature. The present methods for MC, MI and MZalways guarantee the accurate results for a given set of datapoints and do not require any initial solution andconvergence criteria. These algorithms are computationallyquite robust giving unique solutions and require a shortertime for execution. In all cases tested, the results areobtained within a second. The MZ evaluation done usingproposed computational geometric techniques strictlyfollows ISO standards.

The proposed computational geometric techniques forcircularity evaluation fulfill the need for fast and efficientalgorithms for processing data obtained from roundnessmeasuring instruments and practitioners can easily imple-ment them in computer-aided roundness measuring instru-ments for ready industrial applications. The presentmethods also have an advantage that they can be extendedto cylindricity evaluation.

Appendix A. Least squares method for fitting a limacon

The deviation of a point (ri, yi) on the measured profilefrom the limacon represented as ro þ xo cos yi þ yo sin yi isgiven by

ei ¼ ri � ðro þ xo cos yi þ yo sin yiÞ. (1)

On minimizing s ¼P

e2i w.r.t. ro; xo and yo and simpli-fying the normal equations considering equi-spaced andsymmetric data points, the following least squares solutionis obtained:

ro ¼1

N

Xri,

xo ¼2

N

Xri cos yi,

yo ¼2

N

Xri sin yi, (2)

where N represents the total number of measured points.

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