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Vijay SrinivasanReflections on the role of science in the evolution of dimensioning and tolerancing standards

  

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Original Article

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Reflections on the role of science in theevolution of dimensioning andtolerancing standards

Vijay Srinivasan

AbstractDimensioning and tolerancing standards originated about 75 years ago in the form of various national and company stan-dards that governed engineering draughting and documentation practices. They served the purpose of communicating tomanufacturers what geometric variations designers could tolerate in a product without compromising the product’sintended function. These standards have evolved over time and are by now well entrenched in the engineering profes-sion throughout the world. For several initial decades, this evolution was driven primarily by codification of best engi-neering practices without the benefit of any systematic scientific treatment. This trend encountered a major hurdle inearly 1980s when the emergence of computer-aided design and manufacturing systems forced a drastic reexamination ofthese standards with a greater emphasis on mathematical formalism. Since then, scientific principles to explain past prac-tices and to guide future evolution have emerged, and the role of science has now become more prominent in the devel-opment of these standards. This article describes some of the key scientific research results that have already had animpact, and the future scientific trends that are likely to have an influence, on these evolving standards.

KeywordsDimensioning, tolerancing, standards, scientific developments, classification, theory, theorems

Date received: 16 July 2012; accepted: 18 September 2012

Introduction

In 1991, the Pennsylvania State University in theUnited States hosted a College International pour laRecherche en Productique (CIRP) Computer-AidedTolerancing (CAT) Working Seminar. By some currentcount, it was the second of a series of what has come tobe called the CIRP CAT Conferences (the first was agathering on this topic at Jerusalem, Israel, in 1989).The Penn State ‘‘working seminar’’ was a timely eventbecause several academic and industrial researchershad started working together to attack an importantindustrial problem using the computational powerunleashed by the information age.1–4 The author wasfortunate to be present at that workshop and spokeabout how ‘‘a geometer grapples with tolerancing stan-dards.’’5 That workshop, and other similar events orga-nized around that time, launched a series of initiativesthat resulted in the creation of the American Society ofMechanical Engineers (ASME) Y14.5.1 subcommitteeand the International Organization for Standardization(ISO) Technical Committee 213, both dealing with tol-erancing standards. And they spurred one of the most

creative research activities in mechanical and computa-tional sciences, as chronicled in the proceedings of the12th CIRP CAT Conferences and other similar confer-ences and research journals.

Looking back at the article5 written 22 years ago,two of its observations strike us as most consequential.The first was a call for mathematically sound defini-tions of the semantics of the standardized tolerancinglanguage because the lack of such a scientific basis washampering the development of provably correct algo-rithms for CAT. The second was a tentative mention of‘‘computational metrology’’ to refer to a set of compu-tational techniques that were emerging to cope withprocessing large amounts of measured data coming out

Engineering Laboratory, National Institute of Standards and Technology,

Gaithersburg, MD, USA

Corresponding author:

Vijay Srinivasan, Engineering Laboratory, National Institute of Standards

and Technology, 100 Bureau Drive, Gaithersburg, MD 20899, USA.

Email: vijay.srinivasan@nist.gov

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of coordinate measuring machines. It was not clear atthat time if these were merely naı̈ve observations thatwould be promptly forgotten or important topics thatwould be pursued with vigor. In fact, one prominentacademic at the conference scoffed at the very idea ofreferring to tolerancing standards as defining a ‘‘lan-guage with syntax and semantics.’’

It is now heartening to reflect on the developmentsover the past two decades and see that both theseobservations have come to play an important role inthe evolution of dimensioning and tolerancing stan-dards. This article describes some of the scientific devel-opments first (in sections ‘‘Congruence theorems’’through ‘‘Computational coordinate metrology’’) andtheir impact on the evolution of dimensioning and tol-erancing standards next (in sections ‘‘Size tolerancing’’through ‘‘Geometric characteristics and tolerancing’’).Along the way, it will show how these scientific founda-tions have helped rationalize the past and currentindustrial practices, and how they are paving the wayfor important new avenues. Above all, the authorhopes to communicate the excitement of one who haspursued geometric studies for fun as well as profit.

Congruence theorems

Congruence theorems dating back to Euclid provide aneasy and powerful introduction to the notion of dimen-sioning (and parameterizing) geometric objects. Toillustrate, let us start with the simple task of dimension-ing and parameterizing triangles. Figure 1 shows somesuccessful attempts. It seems intuitively obvious that allthese three schemes are valid ways to parameterize tri-angles, and we get valid dimensions when numericalvalues are assigned to the distances and angles indi-cated by arrows.

We can provide a formal theoretical basis for ourintuitive belief by associating each example in Figure 1with a famous triangle congruence theorem fromEuclid’s elements:6 Figure 1(a) with the side-angle-sidetheorem (Book I, Proposition 4), Figure 1(b) with theangle-side-angle theorem (Book I, Proposition 26), andFigure 1(c) with the side-side-side theorem (Book I,Proposition 8).

The geometric notion of congruence is closely relatedto the practical engineering notion of interchangeabilityof parts, as they both belong to an equivalence class.More formally, they satisfy the following three axioms:

1. Reflexivity. A is congruent to (or, interchangeablewith) A.

2. Symmetry. If A is congruent to (or, interchangeablewith) B, then B is congruent to (or, interchangeablewith) A.

3. Transitivity. If A is congruent to (or, interchange-able with) B and B is congruent to (or, inter-changeable with) C, then A is congruent to (or,interchangeable with) C.

In fact, an entire formal theory of dimensioning can bebuilt using congruence theorems.7,8

The taxonomy of such a modern theory of dimen-sioning is shown in Figure 2. At the leaf nodes of themodern taxonomy of dimensioning, we see intrinsicdimensions. At the intermediate nodes, we see relationaldimensions. The hierarchy can be built with as manylevels as the product demands. Complex products mayhave more levels of hierarchy than simpler ones.

Congruence theorems formalize the dimensioningscheme at each level of the dimensional taxonomy. Insection ‘‘Classification of quadrics,’’ we will see howthis is accomplished for some of the commonly usedintrinsic dimensions of surface features trimmed fromquadric surfaces. Then, in section ‘‘Classification ofcontinuous symmetry,’’ we will repeat the exercise forrelational dimensions. As we lay the scientific founda-tion for these dimensioning schemes, it is useful to lookahead to section ‘‘Geometric characteristics and toler-ancing,’’ where tolerancing standards have adopted asimilar hierarchy for ‘‘individual features’’ and ‘‘relatedfeatures’’ invoking the notion of datums. This pointwill be emphasized again in section ‘‘Geometric charac-teristics and tolerancing.’’

Classification of quadrics

Most commonly used surfaces in engineering belong tosecond-degree surfaces called quadrics. These are

(a) (b) (c)

Figure 1. Examples of dimensioning and parameterizing schemes for triangles: (a) side-angle-side theorem, (b) angle-side-angletheorem, and (c) side-side-side theorem.

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defined by a set of points with x, y, and z coordinatesas in�ðx, y, zÞ : Ax2 +By2 +Cz2 +Dxy+Eyz

+Fzx+Gx+Hy+Kz+L=0� ð1Þ

for real coefficients A, B, C, D, E, F, G, H, K, and L,where at least one of A, B, and C is nonzero. A well-known quadric classification theorem9 states that anysurface of second-degree governed by an equation ofthe form (1) can be moved by purely rigid motion inspace, so that its transformed equation can assume oneand only one of 17 canonical forms. Of these 17 cano-nical forms, only 12 can have solutions for real valuesof x, y, and z, and they are shown in Table 1.

The classification theorem also provides a congru-ence theorem: two quadrics are congruent if and only if

they have the same canonical equation. The last col-umn in Table 1 lists the intrinsic parameters of thesesurfaces. The quadrics can be dimensioned by assigningnumerical values to these parameters.

The complete classification of real quadrics plays animportant role in the evolving standardized definitionof ‘‘features of size.’’ Only six of the quadrics enumer-ated in Table 1 belong to one-parameter family of sur-faces, namely, sphere, (right circular) cylinder, (rightcircular) cone, parabolic cylinder, two parallel planes,and two intersecting planes forming a wedge. They alsopossess an important monotonic containment property;for example, a sphere with a larger size dimension con-tains the one with a smaller size dimension. These andother symmetry properties covered in the next sectionprovide the strongest scientific rationale for a standar-dized definition of ‘‘features of size’’ that will be dis-cussed in section ‘‘Size tolerancing.’’

Classification of continuous symmetry

The notion of symmetry greatly simplifies the task ofrelational dimensioning because we then need to proveonly a limited number of congruence theorems. Thesimplification depends on some important classificationtheorems on the connected Lie subgroups of the rigidmotion group and their corollaries on the classificationof continuous symmetry of surfaces.7,8,10,11 These andrelated theorems were rigorously proved only over thepast 15 years.

To gain an intuitive appreciation for the role of sym-metry, consider the problem of positioning an arbitraryobject, such as a chair, in three-dimensional space. Itrequires six dimensions—three for translation and threefor rotation. Now consider positioning a sphere inspace. It seems to require only three dimensions, whichare needed to locate the center of the sphere. We do

Table 1. Classification of real quadrics.

Type Canonical equation Intrinsic parameters

Nondegeneratequadrics

Ellipsoid (x2=a2) + (y2=b2) + (z2=c2) = 1 a, b, and c

Special case: sphere x2 + y2 + z2 = a2 a = radiusHyperboloid of one sheet (x2=a2) + (y2=b2)� (z2=c2) = 1 a, b, and cHyperboloid of two sheets (x2=a2) + (y2=b2)� (z2=c2) = � 1 a, b, and cQuadric cone (x2=a2) + (y2=b2)� (z2=c2) = 0 a/c and b/cSpecial case: right circular cone (x2=a2) + (y2=a2)� (z2=c2) = 0 tan21 (a/c) = semi apex angle

of the coneElliptic paraboloid (x2=a2) + (y2=b2)� 2z = 0 a and bHyperbolic paraboloid (x2=a2)� (y2=b2) + 2z = 0 a and bElliptic cylinder (x2=a2) + (y2=b2) = 1 a, bSpecial case: right circular cylinder x2 + y2 = a2 a = radiusHyperbolic cylinder (x2=a2)� (y2=b2) = 1 a and bParabolic cylinder y2 � 2lx = 0 L

Degeneratequadrics

Parallel planes x2 � a2 = 0 a = half of the distancebetween the parallel planes

Intersecting planes (x2=a2)� (y2=b2) = 0 tan21 (b/a) = half of the anglebetween the intersectingplanes

Coincident planes x2 = 0 None

intrinsic dimensions

intrinsic dimensions

relational dimensions

intrinsic dimensions

relational dimensions

intrinsic dimensions

relational dimensions

intrinsic dimensions

intrinsic dimensions

relational dimensions

Figure 2. A modern taxonomy of dimensioning.

Srinivasan 5

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not need any dimension to specify rotation because thesymmetry of the sphere renders all rotations about itscenter irrelevant for positioning purpose. Finally, con-sider the task of positioning a sphere relative to a(unbounded) plane. A little reflection indicates that weneed to specify only one dimension, namely, the dis-tance between the center of the sphere and the plane.This drastic reduction in the number of needed dimen-sions is due to the fact that the plane also possessessome symmetry because it remains invariant under alltranslations along the plane and all rotations aboutany axis perpendicular to the plane.

Table 2 shows the seven classes of continuous sym-metry. This is a complete classification in the sense thatany surface, in fact any set of points encountered inengineering, belongs to one and only one of these sevenclasses. These are also called invariant classes becausesymmetry is defined by invariance; for example, a cylin-der remains invariant under all translational motionsalong its axis and all rotations about its axis. This istrue about the axis as well because it also remains invar-iant under these motions. There is a powerful theo-rem7,8 that reduces the problem of relative positioningof any two sets to the relative positioning of their simplereplacements, as shown in the last column of Table 2.These results provide the scientific basis for standar-dized datums and datum systems that will be describedin section ‘‘Datums and datum systems.’’

Computational coordinate metrology

Computational coordinate metrology involves thedevelopment and implementation of reliable algorithmsto fit, filter, and perform other types of computationson discrete geometric data collected by coordinate mea-suring systems. Over the past 20 years, it has grown intoa separate research discipline. Several international con-ferences and journals now list computational metrologyas a distinct topic of interest. Of the many researchresults that deal with computational coordinate metrol-ogy, those that address fitting and filtering are the mostrelevant to the verification standards for conformanceto tolerance specifications. Mathematically and compu-tationally, fitting is an optimization problem and mostof filtering is a convolution problem.12–21 The filteringtechniques also form the mathematical basis for surfacetexture characterization.

The scientific and technical advances in hardwarefor coordinate measurements and software for process-ing the data collected from these measurements haveforced a rethinking of tolerance specification standards.To illustrate this fact, let us consider the problem of fit-ting a plane to a set of points in space. If d1, d2, ., dnare the perpendicular (Euclidean) distances of n inputdata points from a plane P, then we can define the dis-tance between this set of points and the plane P usingthe generic lp norm

Xni=1

jdijp( )1=p

ð2Þ

The individual distances can be signed, in the sensethat points lying on one side of the plane can beassigned positive distances and the points lying on theother side can be assigned negative distances. The l1norm is then just the sum of the absolute values of theindividual distances. The l2 norm is the square root ofthe sum of the squares of the distances. The lN norm isthe maximum of the absolute values of the distances; tosee this, we need to look at the definition of the lp normas p tends to infinity. Table 3 presents a set of plane fit-ting problems posed as optimization problems for theobjective function, as shown in equation (2). In the ISOparlance, the l2 norm is known as the Gaussian normand the lN norm is known as the Chebyschev norm.

Algorithms for computing Plane2, PlaneN, andPlaneNC in Table 3 have been well developed in the lit-erature.13 Of these, the least squares plane, designatedas Plane2, is the most widely implemented, tested, andused in industry.22 Plane1C conforms to the ASMEY14.5:2009 standard23 for the establishment of primaryplanar datum because it guarantees a stable plane thattouches at least three points. On the other hand,PlaneNC is the default primary datum plane accordingto the ISO 5459:2011 standard.24 Currently, ISO ismulling over future tolerance specification standardsthat will allow the designer to choose from several fit-ting objectives including the Gaussian norm and theChebyschev norm. Such an expansion is also envi-sioned for a wide variety of geometric characteristicsthat will be described in section ‘‘Geometric character-istics and tolerancing.’’

Size tolerancing

Historically, both the ASME and the ISO standardshad recognized only sphere, cylinder, and two opposingparallel planes as the standardized features of size.These have been the only features to which size dimen-sion and tolerance can be assigned; these have alsobeen the only features that can be designated as datumswith material conditions. These practices originatedfrom engineering experience and were not based on anyscientific rationale.

Table 2. Seven classes of continuous symmetry.

Type Simple replacement

1 Spherical Point (center)2 Cylindrical Straight line (axis)3 Planar Plane4 Helical Helix5 Revolute Straight line (axis) and point-on-line6 Prismatic Plane and straight line-on-plane7 General Plane, straight line, and point

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We can now provide a scientific basis for the defini-tion of features of size and, in that process, expandtheir coverage.25–28 ISO has recently adopted a total offive features of size, as shown in Table 4 from one of itsrecent standards. Five of the six one-parameter familyof quadrics discussed in section ‘‘Classification ofquadrics’’ and five of the seven classes of symmetry(invariance classes) discussed in section ‘‘Classificationof continuous symmetry’’ are represented in thisupdated standardized definition of features of size;these features also possess the monotonic containmentproperty discussed in section ‘‘Classification ofquadrics.’’ Parabolic cylinder, which belongs to theprismatic class, is the only one-parameter family ofquadrics that did not make the list—the engineeringcommunity did not consider its use to be sufficientlywidespread to warrant the ‘‘feature of size’’ designation.

Of the five features of size in Table 4, three—namely,cylinder, sphere, and two parallel opposite planes—have linear units as the intrinsic characteristics. Theseare the linear sizes, and the other two features of sizehave angular sizes. In a recently issued ISO 14405-1standard,29 the linear size tolerances for cylinder andtwo parallel opposite planes have been expanded con-siderably. ISO 14405-1 allows tolerances to be specifiedon the following 14 different types of sizes: two-pointsize, local size defined by a sphere, least squares size,maximum inscribed size, minimum circumscribed size,circumferential diameter size, areal diameter size, vol-ume diameter size, maximum size, minimum size, aver-age size, median size, mid-range size, and range of sizes.

Table 5. Primary datums and datum features according to ASME Y14.5-2009 standard23

Feature type Datum and datum feature simulator

Planar Plane

Width Center plane

Spherical Point

Cylindrical Axis

Conical Axis and point

Linear extruded shape Axis and center plane

Complex Axis, point, and center plane

+0,030-0,040Ø 35 GG

Figure 3. Least squares size tolerancing of a cylinder accordingto ISO 14405-1:2010.29 Here, the GG modifier denotes that aGaussian (least squares) size is being toleranced for theindicated cylinder.

Table 3. Fitting a plane P to a set of points in space.

Objective Constraints Comments Designation

Minimize l1 norm Points lie to one side of P Minimum three points touching P Plane1CMinimize l2 norm None Least squares plane Plane2

Points lie to one side of P Constrained least squares plane Plane2CMinimize lN norm None Minimax plane PlaneN

Points lie to one side of P Constrained minimax plane PlaneNC

Table 4. Features of size according to ISO 5459:201124.

Feature of size Invariance class Intrinsic characteristic

Cylinder Cylindrical DiameterSphere Spherical DiameterTwo parallelopposite planes

Planar Distance between thetwo planes

Cone Revolute AngleWedge Prismatic Angle

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Figure 3 illustrates one of the 14 ways in which the sizetolerance for a cylindrical feature can be specified.

The release of ISO 14405-1 standard in 2010 her-alded a much needed revolution in the tolerance specifi-cation standards. Before the arrival of ISO 14405-1,there was an implicit expectation that size tolerancesshould be verifiable using the likes of micrometers,calipers, dial indicators, and functional gauges. (In fact,such expectations are prevalent even for all geometrictolerance specifications under the guise of ‘‘open set-ups’’ for their verifications.) Using a computerizedcoordinate measuring system was deemed acceptable,as long as it could be run in the ‘‘caliper mode’’ andwith some soft-gauging capabilities. In contrast, a sizetolerance specification such as the one shown in Figure3 can only be verified by coordinate measuring systemsemploying least squares fitting algorithms, which arenow available, thanks to scientific and technical devel-opments in computational coordinate metrologyalluded to in section ‘‘Computational coordinatemetrology,’’ for cylinder fitting. Future ISO standardsfor geometric tolerance specifications will depend criti-cally upon such fitting algorithms for their verification.

Datums and datum systems

Datums and datum systems are essential for specifyingtolerable variations in the relative positioning of fea-tures. As we saw in section ‘‘Classification of continu-ous symmetry,’’ classification of continuous symmetryprovides a scientific basis to define datums. This hasbeen seized upon by the standards community in recenttimes, as briefly described in Table 5 excerpted fromthe ASME Y14.5 standard issued in 2009 and in Table6 excerpted from the ISO 5459 standard issued in 2011.These tables clearly show the classification of standar-dized datums on the basis of the symmetry group clas-sification of the nominal ideal geometric features thatare designated by designers as datum features.

A closer examination of Tables 5 and 6 reveals thedistinction between a datum feature and the associateddatum. For example, a cylindrical feature can be desig-nated as a datum feature, and its axis then becomes thedatum. On an actual workpiece, the features have noni-deal forms and so we need to fit ideal geometric featuresbefore the datums can be discerned. As described in sec-tion ‘‘Computational coordinate metrology,’’ various

Table 6. Invariance classes to define datums according to ISO 5459:2011standard24

Invariance class Situation features Example surface Illustration of examples

Spherical Point Sphere

Planar Plane Plane

Cylindrical Straight line Right circular cylinder

Helical Straight linea Helical thread

Revolute Straight line and point Right circular cone

Prismatic Plane and straight line Elliptical cylinder

Complex Plane, straight line, and point Arbitrary Bezier surface

aHelical surfaces as such are not considered in this International Standard. Natively, the situation feature of a feature belonging to a helical invariance class is a helix, but this International Standard considers only its axis for practical engineering reasons.

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norms can be used to define the objective function inthe optimization problem involved in such fittings. Thestandards currently rely upon the l1 norm and the lNnorm, with constraints; ISO is contemplating appropri-ate symbology to override the default, so that other lpnorms can be invoked for determining the datums.

Geometric characteristics and tolerancing

The dimensional taxonomy shown in Figure 2 was use-ful in structuring the theoretical development of dimen-sioning and geometric parameterization. It alsoexplains the classification of geometric characteristicsand tolerancing as shown in Tables 7 and 8 in theASME and ISO standards, respectively, which havestood the test of time. The ASME classification, asshown in Table 7, refers to individual features andrelated features in the same way intrinsic dimensionsand relational dimensions are dealt with in sections‘‘Congruence theorems,’’ ‘‘Classification of quadrics,’’and ‘‘Classification of continuous symmetry.’’ The ISOclassification, as shown in Table 8, is the same as thatof ASME, and it explicitly invokes the need for datumsfor tolerancing relative positioning.

Both the ASME and ISO tolerancing semantics ofgeometric tolerances shown in Tables 7 and 8 are basedon tolerance zones. ISO is now considering several waysto expand the syntax and semantics of geometric toler-ances as it did for size tolerancing. For example, in thefuture, flatness tolerance can be specified to limit the

root mean square deviation (i.e. standard deviation) ofthe points on an actual feature from a Gaussian (leastsquares) plane fitted to the feature.

Summary

This article described the role of science in classifyingand rationalizing the some of the past and currentdimensioning and tolerancing practices and in pavingthe way for future development of dimensioning andtolerancing standards. Motivated by the industrialneed, the last two decades have witnessed considerablemathematical and algorithmic advances, and thedimensioning and tolerancing standards are well poisedto exploit these advances in at least four distinct areas.

First, computer-aided dimensioning and CAT soft-ware systems can be based on data models that are pro-vably complete and algorithms that are provablycorrect. Second, these data models can form the basisfor standardized exchanges31 that enable interoperabil-ity among engineering information systems. Third,computer-aided manufacturing systems can consumethe tolerancing information automatically for smarternumerical control of machine tools. Fourth, computer-aided inspection systems can use the tolerancing

Table 8. Geometric characteristics according to ISO 1101:2012 standard30

Tolerances Characteristics Datum needed

Symbols

Form Straightness NoFlatness No

Roundness No

Cylindricity No

Profile any line NoProfile any surface No

Orientation Parallelism Yes

Perpendicularity Yes

Angularity Yes

Profile any line YesProfile any surface Yes

Location Position Yes or no

Concentricity (for center points)

Yes

Coaxiality (for axes) Yes

Symmetry YesProfile any line YesProfile any surface Yes

Run-out Circular run-out Yes

Total run-out Yes

Table 7. Classification of geometric characteristics according to ASME Y14.5-2009 standard23

Application Type of tolerance

Characteristic Symbol

Individual features Form StraightnessFlatness

Circularity

Cylindricity

Individual or related features

Profile Profile of a line

Related features Profile of a surface

Orientation Angularity

Perpendicularity

Parallelism

Location Positiona

Concentricity

SymmetryRun-out Circular run-outb

Total run-outb

aMay be related or unrelated.bArrow heads may be filled or unfilled.

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information to generate and execute inspection plansautomatically.

In the next few years, we are likely to witness majorexpansions in the ISO tolerancing standards,32 assistedby scientific developments similar to those outlined inthis article and subsequent codification of concepts andterminology.33–35 Industry will struggle with the magni-tude of such changes, and there will be a great demandfor education of the industrial workforce and collegestudents in these new standards and practices. Wemight well look forward to another two decades of funand profit.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

Acknowledgements

The author would like to acknowledge the contributionsof various subject matter experts in the ASME and ISOstandards committees to the developments reported inthis article. Any mention of commercial products withinthis article is for information only; it does not imply rec-ommendation or endorsement by NIST.

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