A unified approach to form error evaluation

Precision EngineeringJournal of the International Societies for Precision Engineering and Nanotechnology

26 (2002) 269–278

A unified approach to form error evaluation

Timothy Webera, Saeid Motavallib, Behrooz Fallahic,∗, S. Hossein Cheraghid

a Department of Industrial Engineering, Northern Illinois University, Dekalb, IL 60115, USAb Department of Engineering, California State University Hayward, Hayward, CA, USA

c Department of Mechanical Engineering, Northern Illinois University, Dekalb, IL 60115, USAd Department of Industrial and Manufacturing Engineering, Wichita State University, Wichita, KS, USA

Received 1 October 2000; received in revised form 6 September 2001; accepted 24 September 2001

Abstract

Evaluation of form error is a critical aspect of many manufacturing processes. Machines such as the coordinate measuring machine(CMM) often employ the technique of the least squares form fitting algorithms. While based on sound mathematical principles, it is wellknown that the method of least squares often overestimates the tolerance zone, causing good parts to be rejected. Many methods have beenproposed in efforts to improve upon results obtained via least squares, including those, which result in the minimum zone tolerance value.However, these methods are mathematically complex and often computationally slow for cases where a large number of data points are tobe evaluated. Extensive amount of data is generated where measurement equipment such as laser scanners are used for inspection, as wellas in reverse engineering applications.

In this report, a unified linear approximation technique is introduced for use in evaluating the forms of straightness, flatness, circularity,and cylindricity. Non-linear equation for each form is linearized using Taylor expansion, then solved as a linear program using softwarewritten in C++ language. Examples are taken from the literature as well as from data collected on a coordinate measuring machine forcomparison with least squares and minimum zone results. For all examples, the new formulations are found to equal or better than the leastsquares results and provide a good approximation to the minimum zone tolerance. © 2002 Elsevier Science Inc. All rights reserved.

Keywords: Error evaluation; Tolerance specifications; Functional equivalence

1. Introduction

Evaluation of form errors in manufactured parts is essen-tial in determining conformance to tolerance specifications.These specifications are established to provide acceptablelimits on part variation in order to ensure functional equiv-alence. Types of geometric forms typically encountered arestraightness, flatness, circularity, and cylindricity. Becausea surface contains an infinite number of points, in practicethe form must be evaluated using a set of sample points rep-resentative of the surface. The assumption is that this dataadequately represents the surface. The mathematical defini-tions of geometric tolerance zones are given in Appendix A.

Although many algorithms for the evaluation of formtolerance exist, the least squares method is commonlyemployed in equipment such as the coordinate measuringmachine (CMM) for such evaluation. This method is

∗ Corresponding author. Tel.:+1-815-753-9964;fax: +1-815-753-0416.

E-mail address: [email protected] (B. Fallahi).

relatively easy to implement and computation time is min-imal, making it attractive for practical use. However, it iswell known that least squares, in which the sum of thesquares of the deviations from the ideal feature are mini-mized, does not provide the minimum zone result and oftenoverestimates the tolerance zone. Therefore, the use of theleast squares algorithm can result in the rejection of partsthat are within tolerance specifications.

A variety of techniques have been developed which im-prove upon the least squares method, many of which providethe minimum tolerance zone result. However, these meth-ods are mathematically complex and often computationallyslow for cases where a large number of data points are to beevaluated. These cases have become more common with theapplication of measurement devices such as laser scannerswhere they can generate thousands of data points from thepart surface. Also reverse engineering applications where aCAD model for an existing part is generated and evaluatedinvolves evaluation of large amount of surface data. An ex-ample of this case can be found in the inspection of moldsurfaces.

0141-6359/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.PII: S0141-6359(02)00105-8

270 T. Weber et al. / Precision Engineering 26 (2002) 269–278

This report is an effort to improve upon results obtainedfrom the least squares method while keeping complexityand calculation time to a minimum. The non-linear formu-lations of straightness, flatness, circularity, and cylindricityerrors are linearized using Taylor series expansion. In thecase of straightness and flatness, data points are rotated andtranslated to bring the tolerance zone parallel to thex-axisfor straightness andxy-plane in the case of flatness as de-scribed by Cheraghi et al. [3] Taylor expansion is used tolinearize the equations, and the error is then measured inthe positivey (straightness) orz (flatness) directions. Circu-larity error evaluation problem is similarly formulated as anon-linear problem, which is also linearized with Taylor se-ries expansion and solved as linear programming problems.Cylindricity error is evaluated in a set of cross-sections alongthe length of the cylinder. Equations for each cross-sectionare then evaluated as in circularity problems. Though werealize that the result of this algorithm does not provide op-timal value of the cylindricity error based on the minimumradial separation (MRS) criterion, but it is used as a goodapproximation to it.

2. Literature review

The least squares form-fitting algorithms are well knownand thoroughly discussed in Forbes [1] for the forms of lines,planes, circles, spheres, cylinders, and cones, and are notdetailed here. Carr and Ferreira [2] formulate straightnessand flatness as non-linear problems, which are then trans-formed into a series of linear problems that converge to thenon-linear solution points in the set. The efficiency of thesealgorithms with respect to computation time is reported asa linear function of the number of data points. These algo-rithms, as previously mentioned, do provide the minimumzone tolerance values yet require translation into the firstquadrant.

Straightness and flatness have also been formulated asnon-linear problems by Cheraghi et al. [3]. In the straight-ness formulation data points are rotated and translated so theone of the two lines defining the tolerance zone becomes thex-axis. The tolerance zone is then defined by the maximumy-coordinate among the data points. The result is a linearprogram with 2n constraints (n number of data points in theset), 2n−2 of which are redundant. The flatness problem isan extension of straightness with an additional dimension.Data points are rotated through anglesθ andγ about thex-andy-axes, respectively, and translated byX, Y, andZ alongthe x-, y-, andz-axes so that the tolerance zone is definedin the positivez-direction. Exact values of straightness andflatness are obtained using this method, and computationtime is reportedly less than that for other minimum zonealgorithms.

An alternate method of finding the minimum zone tol-erance for straightness and flatness is the convex hull asdiscussed by Traband et al. [4] The convex hull is defined as

the smallest convex set containing all data points. Once theconvex hull is defined around the data points, the methodchecks by iteration each pair of anti-nodal data points. Par-allel lines passing through the points and encompassing alldata points define the tolerance zone (Tf ). The minimumvalue of the distance between these parallel lines for anygiven pair of points is the minimum zone tolerance value.While convex hull minimum zone calculation is complexrelative to other methods, the calculation of convex hullvia anti-nodal pairs reduces the complexity, and thus, com-putation time, for the algorithm. Still, complexity for thistechnique is O(n2) at best.

Shunmugam [5] proposed the minimum average deviationmethod for form error evaluation. This method defines aform such that the deviations of the actual form to this formin either direction are equal, and the tolerance value takenas the maximum peak to valley distance of the deviations.Whereas the least squares method minimizes the sum of thesquares of the deviations (ei), the proposed algorithm seeksto minimize the average deviation, i.e.,

min∑

|ei |

Analogous formulations are given for flatness, circular-ity, cylindricity, and sphericity. The simplex search methodis used to find the minimum average deviation form. Themethod eliminates the weights assigned to the different or-dinate values due to the squaring of the error terms in theleast squares formulations. Examples were carried out andfound, in all cases, to provide smaller tolerance zone valuesthan the method of least squares. However, this techniquedoes not guarantee the minimum zone result.

Lin and Varghese [6] describe the min max algorithm inwhich the maximum value of the deviation (di) from theideal form is minimized. Each data point generates a con-straint. Minimum average deviation, as previously discussed,is also examined in this work. Both formulations are, ofcourse, non-linear, and both arrive at solutions comparableto minimum zone in the tested examples. Neither of thesemethods guarantee the minimum tolerance zone.

Among the many algorithms proposed for circularityevaluation are Voronoi diagrams, a geometric approach tothe minimum zone center as described by Novaski and Bar-czak [7] and Huang [8]. The Voronoi diagram is the resultof drawing perpendicular bisectors between all data points.The nearest Voronoi diagram for a given data point con-tains the region nearest to that point than any otherpoint.

Conversely, the furthest diagram contains the area furtherfrom that point than any other in the set. The edges of thediagrams define unique circle centers from which concen-tric circles are defined about the dataset, beginning with acircumscribed circle around the set or one inscribed insideit. An iterative search is then performed about each vertex,each of which is taken as a new circle center. Concen-tric, MRS circles are then determined based on this center.

T. Weber et al. / Precision Engineering 26 (2002) 269–278 271

The vertex yielding the smallest radial separation then re-sults in the minimum zone tolerance value. Complexity forVoronoi diagram evaluation is reported as O(n2).

Anand and Rajagopal [9] introduced the selective datapartition method for circularity evaluation. This is an it-erative method in which data points are partitioned intoquadrants before fitting concentric circles to the data usingthree different methods. In the first, the mean center for thedataset is used as a starting center coordinate. Horizontaland vertical lines are drawn which divide the dataset intoquadrants. The nearest and furthest points from this cen-ter are located and the respective quadrants noted. The nextnearest point is then chosen from the quadrant 90◦ awayfrom that containing the first closest point and the next fur-thest point selected from the quadrant 90◦ from that forthe first furthest point. Concentric circles are drawn aboutthese points, defining a new center. The procedure is re-peated until points lying on the inner and outer circles al-ternate during a circular scan about the center and all datapoints lie either on the circles or in the annulus. In the sec-ond method, the three furthest points from the mean centerare chosen and an outer circle drawn about them. The in-ner, concentric circle is then drawn about the point lyingclosest to the center. The method terminates when all pointslie inside the outer circle. The third model is based on thedetermination of a circle based on the three closest pointsto the mean center, and a concentric circle is then drawnabout it which encompasses all data points. The processterminates when all points lie between the two concentriccircles. Circularity tolerance is then taken to be the MRSfound among the three methods. Selective data partition re-sults are reported to be better than those obtained from theleast squares and equivalent to those obtained with Voronoidiagrams.

Wang et al. [10] formulate the circularity problem as aMRS problem. They make use of Voronoi diagrams to arriveat necessary and sufficient condition for optimal solution tothe formulated problem. Based on the derived conditionsthey present an efficient algorithm to solve the circularityerror evaluation problem.

Circularity error can be evaluated via maximum inscribedcircle (MIC) or minimum circumscribed circle (MCC). MICis defined as the largest circle which can be inscribed insidethe dataset. Finding the smallest concentric circle to the in-scribed circle which encompasses all data points and takingthe radial separation between the circles results in the MICtolerance. The MIC formulation is as follows:

max min√

[(x − xi)2 + (y − yi)2]

where√

[(x − xi)2 + (y − yi)2] is the radius of the largestcircle which can be inscribed inside the dataset.

MCC is defined as the smallest circle which can be cir-cumscribed around the dataset. The radial separation be-tween this circle and the largest circle concentric to this cir-cle which can be inscribed inside the dataset gives the MCC

tolerance. MCC formulation is as follows:

min max√

[(x − xi)2 + (y − yi)2]

where√

[(x − xi)2 + (y − yi)2] is the radius of the smallestcircle circumscribed around the dataset.

Both MIC and MCC are recommended by ANSI [11] andreported to be relatively fast in terms of computation time,however, neither assures minimum zone results.

Cheraghi and Wang [12] formulate the circularity prob-lem as a MRS problem. MRS is an effective method of cal-culating minimum zone circularity error and complies withASME GD&T standards [13]. Of course the formulation re-quires the solution to a non-linear problem.

The minimum zone cylindricity tolerance, as illustrated byOrady et al. [14] is defined as the volume between two con-centric cylinders containing all data points and satisfying:

|T × (Pi − L)| ≤ t

2

whereT is the direction vector of the cylinder axis,Pi thevector corresponding to the data point (xi , yi , zi), L theposition vector locating the cylinder axis, andt is the radialseparation between the two cylinders. The problem to solveis non-linear with a discontinuous objective function. Theradius of a reference cylinder is adjusted so that the distanceof each data point to the reference cylinder is minimized. Toaid in the convergence of the search procedure and minimizethe possibility of reporting local optima as global solutions,the axis of the reference cylinder is expressed with twopoints rather than the position and direction vectors. Addi-tionally, a synthetic search method is introduced where, asopposed to single-locus searching, multiple loci are exam-ined near the global optimum and the one minimizing theobjective function is chosen. While relatively complex, thisalgorithm is shown to improve upon results obtained vialeast squares and does meet ASME accuracy standards.

The minimum zone formulation is also discussed by Carrand Ferreira [15], including a similar formulation whichsearches for a reference axis rather than cylinder (radiusand axis). In this work, incremental variables are used tocalculate incremental variable changes while searching forthe minimum objective function value. The result is a seriesof linear programs, each created from the previous result,which converge to the non-linear solution.

Carr and Ferreira also examine minimum circumscribedcylinder, which involves finding the minimum cylinderwhich encloses all data points, finding another inscribedinside the data concentric to this cylinder, and taking thecylindricity as the radial separation between the two. Themethod involves the removal ofr1 from the above formula-tion and linearizing the result via incremental variables. Inan analogous fashion, maximum inscribed cylinder is alsoperformed. Examples with these methods result in cylin-dricity values as good as or better than those achieved fromleast squares.


3. Problem formulation

3.1. Straightness

In the least squares method, a least squares line is fit tothe data to minimize the sum of the squared deviations ofdata points from the line. Let (xi , yi) be the coordinates ofa sample point. The least squares line is of the form:

y = ax + c

wherea = (N∑

xiyi −∑xi

∑yi

)/N

∑x2i − (∑

xi

)2,c = ∑

yi − a∑

xi/N andN is the number of data pointsin the set. The orthogonal distance of a given sample pointfrom the least squares line is given by

di = yi − (axi + c)√(1 + a2)

and the least squares straightness tolerance:

max(di ) − min(di )

Based on minimum zone criterion, straightness error-based is defined as the minimum distance between twoparallel lines containing all data elements. To calculatethe minimum zone straightness error, it is assumed thatthe tolerance zone is parallel to thex-axis with thex-axis(y = 0) as one of the two lines defining the zone. Withthis arrangement,y = T s defines the other boundary of theminimum zone whereTs defines the straightness error. Thedata points are rotated through an angleθ and translatedin the y-direction by y0 so that they fall within the zonewhile minimizing the zone width (Ts). The formulation asproposed by Cheraghi et al. [3] is as follows:

Minimize Ts, such that

Y = xi sinθ + yi cosθ + y0 ≤ Ts ≥ 0

The equations are then linearized via Taylor expansion,resulting in the following LP formulation:

Minimize Ts, such that

(xi cosθ0 − yi sinθ0)dθ + dy0 + xi sinθ0

+yi cosθ0 + y0 ≤ Ts ≥ 0

All variables unrestricted.Variables of the LP are dθ and dy0. Theθ0 represents the

angle that are used to generate the linear approximation ofthe constraints. Values of the constants are taken from theleast squares values ofa andc as illustrated in Fig. 1 andshown as follows:

θ0 = −tan−1 a, y0 = −c

3.2. Flatness

The least squares method for flatness case fits a leastsquares plane to the data of the form:

z = ax + by + c

Fig. 1. Straightness.

The distance of a given sample point from the least squaresplane is given by

di = zi − axi − byi − c√1 + a2 + b2

The least squares flatness tolerance is then:


For optimization of flatness tolerance as proposed by Cher-aghi et al. [3] data points are rotated about thex- andy-axesand translated byZ0 in order to bring the tolerance planesparallel to thex − y plane.

The tolerance zone is then measured in the positivez-direction. After rotation and translation, thez-coordinatesof the data points become:

Zi = −xi cosθ sinγ + yi sinθ + zi cosθ cosγ + Z0

where (xi , yi , zi) are the original coordinates of the datapoints, Zi the newz-coordinate,θ the rotation around thex-axis,γ the rotation around they-axis,c the degree of trans-lation, andZ0 is an additional variable allowing movementof the points in thez-direction. The tolerance zone is thendetermined by the distance between tolerance planes (Tf ) inthe positivez-direction. Applying Taylor expansion to theabove results in the following LP:

min Tf , such that

(−xi cosθ0 sinγ0 + yi sinθ0 + cosθ0 cosγ0 + Z0)

+(xi sinθ0 sinγ0 + yi cosθ0 − sinθ0 cosγ0)dθ

+(xi cosθ0 cosγ0 − zi cosθ0 sinγ0)dγ + Z0 = Tf

(−xi cosθ0 sinγ0 + yi sinθ0 + zi cosθ0 cosγ0 + Z0)

+(xi sinθ0 sinγ0 + yi cosθ0 − zi sinθ0 cosγ0)dθ

+(xi cosθ0 cosγ0 − zi cosθ0 sinγ0)dγ + dz0 ≥ 0

All variables unrestricted.Variables in the LP are dθ , dγ , and dz0. Theθ0 andγ 0 are

the angles about which the linearization of the constraints isperformed. Values ofθ0, γ 0, andZ0 are related to the leastsquares estimates of the plane parameters,a, b, andc (see


Fig. 2. Flatness.

Fig. 2). Fig. 2 represents the rotation of the normal vectorto the least squares plane.

θ0 = sin−1(b), γ0 = sin−1[

a

cos(θ0)

],

Z0 = −c cos(θ0) cos(γ0)

3.3. Circularity

Circularity is measured as the radial distance between twoconcentric circles separated by the specified tolerance andcontaining all data elements. In the least squares formula-tion, a circle is fit to the data of the form:

(x − a)2 + (y − b)2 = R2

wherea, b, andR are taken from the least squares formu-lation. For equiangular data, the values fora, b, andR aregiven by

a = 2∑ ri cosθi

N, b = 2

∑ ri sinθi

N,

R =∑ ri

N

The deviation of a sample point from the least squares circleis then:

ei = ri − R − xi cosθi − yi sinθi

Intuitively, the values of the variables, given equiangulardata, can be taken to be:

a =∑ xi

N, b =

∑ yi

N, R =

∑ ri

N

The distance of a sample point from the circle is then:

di =√

(xi − a)2 + (yi − b)2 − R

and the least squares circularity tolerance:


For circularity optimization, we wish to find two radii (R, r)which enclose all data points and whose separation (Tc) isminimum. The formulation is as follows:

min T c = R − r, such that

r ≤√

(xi − a)2 + (yi − b)2 ≤ R

The equations are then linearized via Taylor expansion andthe resulting LP is as follows:

min T c = R − r, such that

r ≤ −(

xi − a√(xi − a)2 + (yi − b)2

)da

−(

yi − b√(xi − a)2 + (yi − b)2

)db

+√

(xi − a)2 + (yi − b)2 ≤ R

All variables unrestricted.The values fora andb are taken from the least squares

result. The variables are da, db, R, andr, allowing variationof the circle center in finding the radii.

3.4. Cylindricity

In the least squares formulation of cylindricity, the centeraxis of a cylinder is fit to the data of the form:

x − x0

u= y − y0

v= z − z0

w

When the cylinder axis is parallel to thez axis,z0 = 0 andw = 1, and the normal distance (di) from data point (xi , yi ,zi) to the cylinder of radiusR is given by

di =

(xi − x0 − uzi )2 + (yi − y0 − zi)

2

+(v(xi − x0) − u(yi − y0))2

1 + u2 + v2

1/2

− R

For this report, in order to obtain initial estimates of thecylinder orientation, the cylinder axis is assumed to be ap-proximately parallel to thez-axis. The cylinder is dividedinto sections based onz-coordinate values of the data points.Each section is then approximated as a circle. Equations ofthese circles are then linearized as in the circularity caseand each data point still provides two constraints. However,in this case, we have a separate LP to solve for each cylin-der section. The least squares cylindricity tolerance is thengiven by


where max(di) and min(di) are the maximum and minimumdeviations from all cylinder sections.

4. Implementation

A program written in C++ language was designed tosolve each of the formulations. Data for straightness andflatness evaluation taken from Cheraghi et al. [3] are illus-trated in Tables 1 and 2. Five examples for each form type


Table 1Straightness results

Example LS [5] OTZa LATb

1 2.4010 2.1213 2.12142 0.8877 0.8578 0.85783 0.1706 0.1646 0.16464 0.0054 0.0052 0.00525 0.0015 0.0013 0.0013

a Optimization technique zone.b Linear approximation technique.

are published and results given for the least squares methodas well as the optimization technique zone. The latter valuesare reported as minimum zone. The algorithms introducedin this report were used on these datasets and results shownfor comparative purposes.

It should be noted that tolerance values for some of thesedatasets are quite high and not representative of practicalcases. In all cases the new method results in smaller zonevalues than the least squares technique, and matches the min-imum zone result when tolerances are practical. Because themethod is an approximation technique, which assumes thatvariations in the values of the variables are small, it may de-viate from the minimum zone result when these variationsare relatively high, as evidenced in example 1 for straight-ness, 1 and 2 for flatness.

The circularity formulation was used on the publisheddatasets in Shunmugam [5] and Orady et al. [16], andprovided the minimum zone result as reported in theserespective articles.

For further evaluation of the new technique, datasets werecreated by measuring various objects using a Brown &Sharpe CMM with PC-DMIS software which incorporatesthe least squares algorithm.

These datasets for straightness and flatness were also eval-uated using the Optimization Technique Zone (OTZ) methodintroduced by Cheraghi et al. [3] and are minimum zone val-ues. The results of these evaluations using each techniqueare reported in Tables 3 and 4. Additionally, the circularitydatasets were evaluated using the minimum radial separa-tion technique, the results of which are reported in Table 5.The cylindricity data was evaluated using the least squarestechnique on the CMM as well as the linear approximationtechnique as given in Table 6.

Table 2Flatness results

Example LS OTZa LATb

1 2.3664 1.9612 2.07152 9.1797 4.8573 4.86263 0.1856 0.1465 0.15614 0.0044 0.0042 0.00425 0.0030 0.0026 0.0026


Table 3Straightness


1 0.0017 0.0017 0.00172 0.0015 0.0014 00143 0.0049 0.0047 0.00474 0.0025 0.0023 0.00235 0.0011 0.0011 0.0011


Table 4Flatness


1 0.0002 0.0002 0.00022 0.0028 0.0027 0.00273 0.0002 0.0002 0.00024 0.0033 0.0032 0.00325 0.0026 0.0025 0.0025


Table 5Circularity

Example LS MRSa LATb

1 0.0006 0.0006 0.00062 0.0022 0.0018 0.00183 0.0012 0.0011 0.00114 0.0279 0.0253 0.02535 0.0009 0.0008 0.0008

a Minimum radial separation.b Linear approximation technique.

Table 6Cylindricity

Example CMM LATa

1 0.0067 0.00602 0.0006 0.00053 0.0051 0.00424 0.0004 0.00045 0.0007 0.0004

a Linear approximation technique.

Because these data were collected from actual machinedparts, it is representative of typical tolerance zones, whichmight be evaluated, in industrial applications. This data isprovided in Appendix B. Calculation time for these problemsis typically no more than a second.

5. Conclusions

The results reported in this paper illustrate the effective-ness of the linear approximation technique relative to leastsquares and minimum zone calculations. For all examples


in the study, the new technique matches or improves uponthe least squares result and, in each case for the datasetscollected from the CMM, results in the minimum zone tol-erance value.

It should be noted that reasonable initial estimates of thevariables are required in order to obtain the desired results.From the analysis it is evident that the method of leastsquares provides sufficient initial values. Moreover, whenusing least squares to obtain a starting point, the linear ap-proximation technique cannot result in larger tolerance val-ues because we are seeking to minimize a function havingthe least squares solution as an initial condition. Becausethis first requires the calculation of least squares coefficients,the linear approximation technique naturally requires morecomputation time than least squares. However, in practicethis difference should prove negligible, and the calculationtimes indicated in the previous section appear to supportthis.

From this study, it is evident that the linear approximationtechnique is a viable method of form tolerance evaluation.Solving the linear problems reduces complexity consider-ably relative to non-linear solution techniques, making thenew technique suitable for implementation in equipmentcurrently using least squares for tolerance evaluation.

Appendix A. Standard mathematical definition oftolerance zones

A straightness zone for a surface line element is an areabetween parallel lines consisting of all points�P satisfyingthe condition:

|T̂ × ( �P − �A)| ≤ t

2

where T̂ is the direction vector of the center line of thestraightness zone;�A a position vector locating the center

Appendix B

CMM-straightness datasets (all data points are in inches):

Example 1 Example 2 Example 3 Example 4 Example 5

x y x y x y x y x y

0.6981 −1.9966 0.0674 0.0009 2.3481 2.2383 2.3786 −1.6979 0.6272 0.42231.0762 −1.9901 0.2251 −0.0002 2.5964 2.2382 2.5497 −1.6972 0.6854 0.42401.4104 −1.9827 0.3761 −0.0004 2.9185 2.2371 2.6900 −1.6963 0.7374 0.42561.8294 −1.9732 0.5463 −0.0002 3.2000 2.2354 2.8297 −1.6956 0.8354 0.42902.2831 −1.9632 0.7011 −0.0002 3.4936 2.2330 2.9489 −1.6946 0.9477 0.43162.7328 −1.9531 0.8730 −0.0002 3.8226 2.2326 3.0709 −1.6937 1.0318 0.43453.1660 −1.9441 0.9773 −0.0001 4.0315 2.2316 3.2127 −1.6929 1.1084 0.43683.5183 −1.9362 1.0734 0.0000 4.1963 2.2313 3.3633 −1.6922 1.1914 0.43963.8147 −1.9294 1.1821 −0.0001 4.4125 2.2308 3.4880 −1.6913 1.2631 0.4417

line of the straightness zone;t the size of the straightnesszone (the separation between the parallel lines).

A flatness zone is a volume consisting of all points allpoints �P satisfying the condition:

|T̂ × ( �P − �A)| ≤ t

2

whereT̂ is the direction vector of the parallel planes definingthe flatness zone;�A a position vector locating the mid-planeof the flatness zone;t the size of the flatness zone (the sep-aration between the parallel planes).

A circularity zone at a given cross-section is an annu-lar area consisting of all points�P satisfying the condi-tions

|T̂ × ( �P − �A)| = 0

and

|| �P − �A| − r| ≤ t

2

whereT̂ for a cylinder or cone, a unit vector that is tangentto the spine at�A. For a sphere,̂T is an unit vector thatpoints radially in all directions from�A. �A is a position vectorlocating a point on the spine;r a radial distance (which mayvary between circular elements) from the spine to the centerof the circularity zone (r > 0 for all circular elements) andt the size of the circularity zone.

A cylindricity zones a volume between two coax-ial cylinders consisting of all points�P satisfying thecondition:

||T̂ × ( �P − �A)| − r| ≤ t

2

where T̂ the direction vector of the cylindricity axis;�A aposition vector locating the cylindricity axis;r a radial dis-tance from the cylindricity axis to the center of the tolerancezone andt the size of the cylindricity zone.


Appendix B (Continued)


x y x y x y x y x y

4.1650 −1.9223 1.2896 0.0003 4.6058 2.2299 3.6903 −1.6896 1.3482 0.44461.4353 0.0002 4.8005 2.2300 3.8789 −1.6897 1.4439 0.44811.5221 0.0003 5.0218 2.2301 4.0420 −1.6892 1.5252 0.45061.6652 0.0002 5.2660 2.2310 4.2596 −1.6855 1.5853 0.45231.7786 0.0003 5.4763 2.2305 4.3979 −1.6861 1.7184 0.45691.8625 0.0006 5.6524 2.2302 4.5625 −1.6856 1.8010 0.45952.0028 0.0006 5.8317 2.2300 1.8685 0.46102.0884 0.0005 6.0326 2.2302 1.9403 0.46282.1836 0.0006 6.2811 2.2303 2.0075 0.46502.2842 0.0008 6.4739 2.2305 2.0789 0.46752.3745 0.0009 6.7640 2.23072.4790 0.00102.6024 0.00112.6805 0.00122.7522 0.00122.8591 0.00112.9650 0.00143.0973 0.00123.2302 0.00153.3711 0.00213.4836 0.00173.6605 0.00123.8265 0.00153.9174 0.00114.0053 0.00144.1246 0.0012

CMM-flatness datasets:


x y z x y z x y z x y z x y z

1.8050 0.8763 −11.2874 −0.2054 2.454 −13.6502 1.9391 2.5884 −13.0151 1.5509 3.1847 −12.6708 1.4922 0.9885 −11.55602.3901 0.8763 −11.2880 0.6986 2.4539 −13.6517 2.7537 2.5884 −13.0161 2.6321 3.1847 −12.6740 1.4922 0.5775 −11.55393.0622 0.8762 −11.2886 1.7277 2.4538 −13.6542 3.7593 2.5884 −13.0173 3.6875 3.1847 −12.6742 1.4922 0.2091 −11.55263.8511 0.8762 −11.2893 3.1273 2.4538 −13.6531 4.3794 2.5883 −13.0181 4.7256 3.1846 −12.6720 1.4922 −0.1810 −11.55173.3626 0.4425 −11.2887 4.2423 2.4537 −13.6557 5.1333 2.5883 −13.0190 4.7257 2.7438 −12.6716 1.4922 −0.6739 −11.55113.3626 0.4425 −11.2883 4.5039 1.5683 −13.6532 5.1334 2.0145 −13.0182 3.9723 2.7439 −12.6729 1.8597 −0.6739 −11.55152.7967 0.4425 −11.2878 3.2475 1.5684 −13.6532 4.4766 2.0145 −13.0176 2.6992 2.7439 −12.6732 1.8597 −0.3101 −11.55182.0039 0.4425 −11.2872 1.9458 1.5684 −13.6519 3.7038 2.0146 −13.0165 1.5277 2.7439 −12.6697 1.8597 0.0708 −11.55241.9860 −0.0969 −11.2865 0.9028 1.5685 −13.6496 2.8800 2.0145 −13.0156 1.5278 1.7473 −12.6687 1.8597 0.4658 −11.55362.6751 −0.0969 −11.2871 −0.1095 1.5686 −13.6489 1.9602 2.0146 −13.0143 2.6923 1.7473 −12.6716 1.8597 0.8731 −11.55543.4853 −0.0970 −11.2878 −0.1095 0.5439 −13.6451 1.9602 1.4227 −13.0135 3.7360 1.7473 −12.6718 2.1079 0.8731 −11.55444.1801 −0.1325 −11.2882 0.8310 0.5438 −13.6462 2.6978 1.4227 −13.0145 4.7501 1.7472 −12.6702 2.1080 0.6291 −11.5532

2.3130 0.5437 −13.6489 3.8093 1.4227 −13.0159 4.7501 0.8951 −12.6701 2.1080 0.2732 −11.55203.4521 0.5437 −13.6501 4.3840 1.4226 −13.0166 3.7441 0.8952 −12.6709 2.1081 −0.0488 −11.55134.5851 0.5436 −13.6509 4.8801 1.4226 −13.0172 2.7306 0.8970 −12.6706 2.1081 −0.4016 −11.55084.5851 −0.3540 −13.6485 1.6833 0.8970 −12.6683 2.3771 −0.4017 −11.54893.7475 −0.3539 −13.6492 1.6834 0.0752 −12.6677 2.3847 −0.1794 −11.54912.4840 −0.3539 −13.6465 2.7906 0.0752 −12.6704 2.4445 0.1066 −11.54901.0032 −0.3538 −13.6468 3.8180 0.0752 −12.6702 2.4627 0.3877 −11.5495

−0.4824 −0.3537 −13.6449 4.9965 0.0751 −12.6693 2.4850 0.7268 −11.5503


CMM-circularity datasets:


x y x y x y x y x y

−0.1690 0.3360 −0.0017 0.2476 −0.1512 0.2608 0.0812 0.2758 −0.1376 0.5963−0.3631 0.1275 −0.1754 0.1905 −0.2890 0.1094 −0.1308 0.2308 −0.2504 0.5350−0.3561 −0.1533 −0.2649 0.0676 −0.2615 −0.1691 −0.2745 0.1038 −0.3624 0.3896−0.1233 −0.3620 −0.2748 −0.0807 −0.1517 −0.2701 −0.3318 −0.0706 −0.3865 0.1762

0.1925 −0.3177 −0.2409 −0.1658 −0.0096 −0.3050 −0.3193 −0.1956 −0.3455 0.06030.3494 −0.0985 −0.1980 −0.2188 0.1820 −0.2328 −0.2798 −0.2890 −0.2083 −0.08100.3222 0.1613 0.0379 −0.2910 0.2786 −0.0783 −0.2030 −0.3778 0.0765 −0.12120.1413 0.3364 0.1767 −0.2190 0.2799 0.0667 −0.0018 −0.4634 0.2180 −0.0496

0.2461 −0.1078 0.2127 0.1940 0.1739 −0.4126 0.3264 0.09420.2449 0.0687 0.0659 0.2840 0.2842−0.3452 0.3234 0.39320.1974 0.1518 0.3477 −0.2915 0.2773 0.47030.1348 0.2064 0.4073 −0.1371 0.1810 0.5577

0.3995 0.01480.3545 0.10960.2869 0.18600.1386 0.2477

CMM-cylindricity datasets:


x y z x y z x y z x y z x y z

−0.0736 0.1509 −7.2331 −0.0836 0.4884 −11.9559 −0.1278 0.5086 −10.1993 0.0066 0.2427 −9.4313 −0.1303 0.4696 −13.2956−0.1632 0.0739 −7.2332 −0.4842 0.1162 −11.9556 −0.5316 0.0963 −10.1985 −0.1902 0.1045 −9.4311 −0.4371 0.2102 −13.2963−0.1366 −0.0808 −7.2559 −0.1825 −0.4606 −11.9555 −0.4447 −0.3252 −10.1984 −0.1141 −0.1210 −9.4352 −0.2859 −0.4115 −13.2963−0.0097 −0.1381 −7.2583 0.3051 −0.3862 −11.9556 −0.0509 −0.5516 −10.2061 0.0146 −0.1564 −9.4351 0.2903 −0.4261 −13.2966

0.0510 −0.1181 −7.2584 0.4249 0.2472 −11.9558 0.3719 −0.3926 −10.2072 0.1483 −0.0911 −9.4353 0.5134 −0.0079 −13.29670.1122 −0.0374 −7.2584 0.0954 0.4850 −11.9558 0.5243 −0.0353 −10.2079 0.1964 0.0816 −9.4357 0.2443 0.4338 −13.29670.0940 0.1076 −7.2596 −0.0440 0.4937 −11.7604 0.4409 0.2719 −10.2080 0.0945 0.2193 −9.4369 −0.1119 0.4751 −13.11130.0084 0.1601 −7.2629 −0.4732 0.1548 −11.7601 0.1854 0.4801 −10.2076 −0.0446 0.2377 −9.4370 −0.4261 0.2301 −13.1111

−0.1106 0.1339 −6.9202 −0.2093 −0.4487 −11.7600 −0.2699 0.4535 −9.8828 0.0062 0.2415 −9.2827 −0.2596 −0.4289 −13.1111−0.1687 0.0614 −6.9203 0.3246 −0.3699 −11.7601 −0.5388 0.0981 −9.8888 −0.1570 0.1655 −9.2825 0.3292 0.3979 −13.1113−0.1387 −0.0804 −6.9204 0.4807 0.1035 −11.7603 −0.4402 −0.3449 −9.8888 −0.1673 −0.0671 −9.2820 0.5136 0.0177 −13.1114−0.0065 −0.1374 −6.9207 0.2118 0.4465 −11.7603 −0.0703 −0.5528 −9.8894 −0.0312 −0.1557 −9.2817 0.2173 0.4475 −13.1113

0.0872 −0.0833 −6.9208 −0.0399 0.4948 −11.5540 0.3772 −0.3838 −9.8906 0.1239 −0.1152 −9.2818 −0.0878 0.4806 −12.90560.1231 0.0394 −6.9208 −0.4479 0.2177 −11.5537 0.5180 0.0182 −9.8912 0.1998 0.0342 −9.2821 −0.4342 0.2135 −12.90550.1107 0.0774 −6.9216 −0.1954 −0.4542 −11.5535 0.3744 0.3470 −9.8912 0.1557 0.1674 −9.2824 −0.3413 −0.3618 −12.90550.0534 0.1425 −6.9240 0.2909 −0.3970 −11.5536 −0.0032 0.5175 −9.8905 0.0029 0.2415 −9.2860 0.2873 −0.4289 −12.9058

0.4727 0.1383 −11.5538 −0.0424 0.2352 −9.0846 0.5122 0.0407 −12.90590.1085 0.4835 −11.5538 −0.1867 0.1114 −9.0864 0.2405 0.4368 −12.9058

−0.1636 −0.0740 −9.08890.0167 −0.1591 −9.10380.1620 −0.0769 −9.10410.1998 0.0282 −9.10440.1580 0.1631 −9.10470.0324 0.2376 −9.1059

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Documents

A unified approach to form error evaluation