GEOMETRIC TOLERANCES: A NEW APPLICATIONFOR LINE GEOMETRY AND SCREWS

J. K. Davidson J. J. Shahand , ProfessorsDepartment of Mechanical and Aerospace Engineering

Arizona State UniversityTempe, Arizona 85287-6106

J.Davidson@asu.edu

Abstract

A new mathematical model is introduced for the tolerances of cylindrical surfaces. Themodel is compatible with the ISO/ANSI/ASME Standard for geometric tolerances.Central to the new model is a Tolerance-Map , a hypothetical volume of points which®1

corresponds to all possible locations and variations of a segment of a line (the axis) whichcan arise from tolerances on size, location, and orientation of the cylindrical surface.Each axis in a tolerance-zone will be represented with the six Plücker coordinates.Cylindrical surfaces in a tolerance-zone for the same hole can then be treated by attachinga size tolerance to each of the lines, thereby forming a screw. Relationships for thecontent of line-solids for a tolerance-zone are developed to correspond to the variationsof locations. These are then used to get a measure for the increment in cost when a morerefined tolerance is specified. This model is one part of a bi-level model that we aredeveloping for geometric tolerances.

1. Introduction

Two types of principles are used in industry for assigning tolerances: conventionaltolerancing and geometric tolerancing. In conventional dimensioning and tolerancing, atolerance that is applied to a dimension is considered to permit one degree of freedom tothe target entity, i.e. displacement only in the direction of the dimension. Geometrictolerancing refers to modern methods that are prescribed in the ANSI/ASME Y14.5MStandard (1994). Here the allowed variation is defined with a in whichtolerance-zone

1Patent pending; provisional patent No. 60/120,961 (1999).

the target entity is permitted several degrees of freedom of displacement. The location ofthe tolerance-zone is constrained with a basic dimension; its size and internalcharacteristics are determined by the tolerance values, such as size, location, andorientation. The Standard allows for interactions between certain tolerances, such asbetween location and size. It also allows for the ordering of the precedence of datumsthat can be used for successive refinements in control and that should be used whenmeasuring the part to determine its acceptability. For example, in Fig. 1 Datum is the�primary datum and Datums and are the secondary and tertiary ones.� �

There are five generic approaches that have been proposed to represent geometrictolerances. They are: parametric models, offset zone models, variational surface models,kinematic models, and degree-of-freedom (DOF) models. We follow with briefcomments about each.

Fig. 1. Two holes in a plate of thickness . Both holes are located with the�

tolerance mm. The larger hole is to be held perpendicular to Datum ! y �»� (

with the tolerance mm.! y �»ZZ

Parametric models: In parametric CAD, the nominal shape and size is represented bya set of explicit dimensions and constraints from which a set of simultaneous equationsare obtained. Solving these equations gives one or more values for the dependentdimensions. Tolerances are incorporated by allowing +/- variations in the dimensions (seee.g. Gossard, . (1988), Hillyard and Braid (1978), and Light and Gossard (1982)).et alThe method has been successful with two-dimensional profiles, but its application to thegeneral three-dimensional problem is limited, in part, because equations are written andsolved for vertex positions, thereby limiting application to polyhedral parts. Formtolerances cannot be included.

Offset zone models: A tolerance zone is modeled as Boolean subtraction of maximaland minimal object volumes that are obtained by offsetting the object by amounts equalto the tolerances on either side (see e.g. Requicha (1983), Requicha and Chan (1986), and

Roy and Liu (1988)). The construction of such a composite tolerance zone fromboundary surfaces of the part does not allow one to model each type of variationseparately, nor to study their interactions.

Variational surfaces: Each surface is varied independently by changing the values ofmodel variables from which surface coefficients are calculated (see Martensen (1993) andTurner (1990)); positions of the vertices and edges are computed from the surfacevariations. Form tolerances can be handled either by using higher degree surfaces thanthe nominal surface or surface triangulation. This model leads to some topologicalproblems, such as maintenance of tangency and incidence conditions. This model, too, asthe previous two described, is incompatible with the ANSI/ASME Standard (1994).

Kinematic models et al: Building on prior work (Rivest, ., 1994) to analyze tolerancestack-up in mechanisms using transformation matrices, Chase . (1998) developed aet alkinematic model by representing each tolerance class by a combination of kinematicjoints. These combinations were then used to estimate geometric variations caused byfeature tolerances. They have yet to show how this approach can be extended to combinethe interaction of geometric variations with size dimensions.

DOF models: It appears that during the time period 1988-1993, several groups (andASME) were independently working on some form of spatial degree of freedom (DOF)model to represent geometric tolerances; these models are forms of the kinematic models.Kramer (1992) used symbolic reasoning to demonstrate the determination of degrees offreedom of parts in an assembly and to determine assembly feasibility based on nominaldimensions. Clément . (1991, 1992) used elementary surfaces (planes, cylinders,et alspheres, etc.) to model the six lower kinematic pairs, as identified by Reuleaux (1875),together with the complete constraint of a fixed rigid body, in a manner described byHunt (1976). They called these the seven technologically and topologically relatedsurfaces (TTRS) from which they elaborated the 28 different geometric relationshipspossible and the consequent remaining degrees of freedom for each combination. A(differential) displacement torsor was defined as a 6 dimensional vector, containing 3rotation values and 3 translation values; the torsor is a set of screw-coordinates scaled toinclude the amount of differential displacement. For each tolerance related to a TTRS thetolerance zone was represented as a torsor containing the non-invariant rotation andtranslations. However, with this representation, one is unable to distinguish betweenvariations resulting from size, form, and location. Also, datum precedence was notconsidered.

A variation on the kinematic approach is to use torsors to combine both the errorsfrom joint-clearances and ones from manufacturing errors. Desrochers and Rivière(1997) present each torsor as a four-by-four matrix. Samper and Giordano (1998) andDeroschers (1999) use torsors directly. Tischler and Samuel (1998) use torsors to modelthe deviations in position for a robot end-effector which arise from accumulated slop atthe joints. With this variation, a target feature is considered to be attached to a rigid body,

and the resultant torsor, taken as the sum of the torsors from the contributing errors,represents the accumulated total of both rotational and translational errors.

The purpose of this paper is to introduce a new role for lines and screws that will beused in a mathematical model for characterizing the tolerances for location andorientation of a line. We further extend the model to cylindrical surfaces that have a sizetolerance on diameter, such as bosses and holes. The new model is distinct because thescrew is used entirely geometrically; it is not used as a vehicle to describe the motion of arigid body. The model is used to create Tolerance-Maps, i.e. point spaces, the shapes andsizes for which differentiate between the variations provided by different tolerances.

2. The Tolerance-Map (T-Map) for a Line®

A Tolerance-Map is a hypothetical -dimensional volume of points, the shape, size, and�internal component-zones of which represent all possible variations in size, form, andorientation of a feature. This point-space is established in Euclidean space withsuperabundant homogeneous coordinates but can also be described with Cartesiancoordinates.

2.1 Areal coordinates. In creating a T-map for the tolerance-zone of a line (e.g. the axisof a hole or a boss), we choose to use areal coordinates. This paragraph sets outimportant definitions and some properties of these coordinates, as taken from Coxeter(1969), Klein (1939), and Sommerville (1929). At five fixed points $ , $ , $ , $ , and $ ,� � � �

called basis points that are arbitrarily chosen in Euclidean space to form a 4-simplex(Coxeter, 1969), we place five masses , , , and that may be positive or� � � � �� � � �� 5

negative (see also the Appendix). So long as 0 ( , , ), the position $ of��� � � � �the centroid of these masses is uniquely determined by the linear combination

� ��� y�

� � � � � � � � � � � � � � �$ $ $ $ $ $ , (1)� � � � �

and we can make it assume any position in the point-space of by varying the five$ $ � 5

� � �� �. The five masses also can be regarded as the of $. 5 barycentric coordinates We note that the position of $ depends only on four independent ratios of thesemagnitudes, thereby establishing 4 as the dimension of the space. When the coordinates� � ��� � � 5 are normalized by setting , they become , a term thatareal coordinatesderives from the two-dimensional array of three weighted points in which each coordinaterepresents a fraction of the area of the reference (or basis) triangle (Coxeter, 1969). Inthis paper, areal coordinates are used to describe T-maps for lines on parts.

2.2 The Tolerance-Map (T-map®). In Fig. 1 is shown a plate of thickness with two holes that are located with the geometric tolerance mm. According to the� � ��ANSI/ASME Y14.5M standard (1994), the range of locations for an axis is represented asin Fig. 2, i.e. all acceptable locations of the axis are identified by the line-segments of

length (Fig. 1) that can be placed in the long thin right-circular cylinder defined by circles and of diameter . One can visualize the possibilities to be the same as those� � �

�

for locating a soda straw of length and infinitesimal diameter in a cylindrical can, also of length , which is bounded at its top and bottom by the very small circles and , � �

�

respectively, each of diameter . Such a set of lines (each of infinite extent) on which the�line-segments lie was by Ghosal and Roth (1987). called a line-solid Now, to apply theconcept of areal coordinates, above, consider that $ , $ , $ , $ , and $ are five � � � � basis-lines that define the space of our four-dimensional set of lines in Fig. 2. We map theselines to five corresponding that we arrange to be the vertices $ of abasis-points � $ 5

simplex in a four-dimensional point-space and we assign one of the basis-lines to each.Setting , any point $ in the point-space of the simplex can be represented as a��� � weighted linear sum of the basis points, i.e.

$ $ $ $ $ $ . (2)� � � � �� � � � �� � � � � � � � 5 5

Fig. 2. A cylindrical tolerance-zone for the axis of a hole determined by circles*T

and , each with its diameter equal to the positional tolerance . Basis-lines* !

$ $ are shown, together with four others that complete the symmetry.� ¾

For , , , , and all positive, $ identifies any point the simplex� � � � �� � � 4 5 inside $ $ $ $ $ . The construction described above implies a one-to-one relationship between� � � � 5

the line-segments in the tolerance-zone of Fig. 2 and points in the four-dimensional spacethat is described with areal coordinates. It further suggests that any line in the tolerance-zone of Fig. 1 can be identified with the linear relationship in eqn (2), a suggestion thatclearly is not valid in general because linear combinations of two lines yield , notscrews lines. However, we use eqn (2) for the lines in a tolerance-zone because every onemay of these lines is rotated only slightly from the theoretical orientation of the feature axis.This assertion will be confirmed in §4.

Since the T-map is a bounded volume in the four-space of points, graphicalrepresentations only can occur with projections or with cross-sections through the whole;three two-dimensional cross-sections are shown in Fig. 3. Since each of Figs. 3( ), ( ),� �and ( ) contains just three of the basis-points, each is a two-dimensional section through�the Tolerance- . However, although it is appealing to also regard these as the -,Map ���� ��-, and -sections, respectively, through the tolerance- , the interpretations of thezone

Fig. 3. Three two-dimensional sections through the Tolerance-Map (fourdimensional point-map) for the tolerance-zone shown in Fig. 2. ( ) The central�

plane . ( ) Plane . ( ) Plane .� � � � � �� �y � y � y � y � y � y3 2

figures are not consistent. Points that are on, or anywhere within, the circle shown in Fig.3( ) correspond to one of the lines that is parallel to the -axis in the tolerance-zone of� �Fig. 2. of these lines lies in the -plane of Fig. 2. Points lying inside the basis-None ��triangle $ $ $ (only two sides are drawn in Fig. 3( )) have only positive values for the� � � �three -coordinates, but points outside require at least one of these to be negative. For�

instance, the areal coordinates for $ are ( ) ( – ) and for $ � � � � � � � � �� � � � � �� � �� �� � 7

are ( – ).�� �� � �� �The interpretation of Figs. 3( ) and 3( ) as corresponding to - and -cross-� � �� ��

sections of the tolerance-zone in Fig. 2 does meet the traditional test: points in each ofthese figures correspond to all the lines that lie the respective planes of Fig. 2. Thein points on the line-segment $ $ in Fig. 3( ) correspond to those lines in Fig. 2 that lie in� � �the -plane, pass through point , and form a partial pencil of lines in the tolerance-�� �zone between $ and $ . Similarly, line-segment $ $ corresponds to point in Fig.� �4 4 �3( ), i.e. those lines in the tolerance-zone of Fig. 2 which form a partial pencil centered at�� �� � and lie in the -plane. Any point $ in Fig. 3( ) can be represented as a weightedlinear sum of the basis-points $ , $ , and $ using eqn (2), and being zero for all� � � � � �

points in the cross-section. For , , and all positive, $ identifies any point inside� � �� � �

� �� �� �� � �$ $ $ ; $ , being outside, takes the coordinates ( – ). All the comments� � � �

concerning Fig. 3( ) also apply to Fig. 3( ), except that $ , $ , and $ are now the basis-� � � �

points, and are zero throughout, and $ and point now take the roles of $ and� �� � ��point , respectively.�

The shape is important for the 4-simplex $ $ $ $ $ used as a basis for the T-map.� � � � 5

In this paper we use a non-regular 4-simplex, a shape such that isosceles triangles withleg-length are formed in each of the three sections of Fig. 3. For example, in Fig.���3( ) $ , $ , and $ form one isosceles right triangle. The square cross-sections in Figs.� � � �

3( ) and 3( ) are each formed from two isosceles triangles set base-to-base which have� �aspect ratio 1 2 (ratio of height- to base-dimension). There are two reasons for choosing�this shape for the 4-simplex. First, a Cartesian frame of coordinates can be overlain onthe simplex and on the Tolerance- that is derived from it, the Cartesian frame beingMaprelated to a subset of the Plücker coordinates ( ) that describe the�� �� � ! � "� #position of a line in the tolerance- ( , , and represent the direction ratios forzone. � � �the line and the vector ( ) represents the moment about of a unit force that acts on! "# �the line.) For example, the coordinates ( ) for line $ in Fig. 2 are�� �� � ! � "� # �

( ), and for $ they are ( ). By choosing the isosceles shapes�� �� ���� �� � �� ��$� �� �� �

in Fig. 2, two of the Cartesian coordinates can be and and the others, which we! "designate and , have dimension [length] and are proportional to the ratios and� � �Z Z

� � � � ! � ", respectively. The overlay of coordinates ( ) provides consistent units in� Z Z

the Euclidean space of the T-map.Second, the isosceles shape simplifies the interpretation of the T-map. For instance,

points in the Tolerance- that are displaced vertically from the central planeMap correspond to lines in the tolerance- that are tilted more away from the vertical.zone

Further, every line-segment in the T-map that is parallel to (or in) the plane of $ , $ , and� �

$ represents a set of parallel lines in the tolerance-zone, and every other line corresponds�

to a partial pencil of lines, i.e. the dual to a line segment of points.There is a danger in presenting in one place the three cross-sections shown in Fig. 3

because it at first appears that they could be superimposed, then making $ and $�

coincide. Such superposition must be resisted, however. The lines $ and $ are the� not same line in the tolerance-zone (Fig. 2) and must be treated as distinct points in theTolerance-Map. Figures 3( ), 3( ) and 3( ) are simply three two-dimensional cross-� � �sections of the T-map, each of which contains three basis-points. Figure 3( ) is the cross-�section that represents the central plane of the T-map.

3. The Coordinates for a Line in the Tolerance-Zone

It is convenient to formulate the Plücker coordinates for a line in the tolerance-zone fromtwo distinct points and through which it passes; the coordinates%� � � � � & %� � � � � &� � � � � �

are (see, e.g. Hunt, 1990) the six 2x2 determinants from the array

� � � � � � � �

�� � �

� � �

So, for the geometry in Fig. 4, the Plücker coordinates of a line in the tolerance-zone'can be obtained from the two points to give the array! and (

� � � � �

Z Z

� �, (3)

in which the values for , , , and are at least two orders of magnitude smaller than� �Z Z � �

��� and 1. The line then is represented in the -frame of Fig. 2 by' ) ( )�� �� � ! � "� # in which the coordinates are

� � * � � � * � � �

! � � � " � � � # � � * � �

� �

� � � �

Z Z

Z Z Z Z� �

� �� � � � �–

(4)

We note that, for line-segment in the tolerance-zone of Figs. 2 and 4, and isevery � � a 0'th order quantity; , , , and are small quantities on the order of the tolerance,� � ! "i.e. [ ]; and is of order [ ] . Therefore, coordinate is negligible, and distinctions� # � #�

between line-segments in the tolerance-zone are made only with , , , and . It is� � ! "for this reason that these are the only four coordinates that are related to the Cartesiancoordinates that overlay the T-map and the cross-sections through it (Fig. 3).

4. Linear Combinations of Lines

Although any line in the tolerance-zone of Figs. 2 and 4 can be obtained by selecting apoint in each of circles and , it is not so clear that eqn (2) represents the same thing.� �

�

For instance, all linear combinations of $ and $ give a cylindroid of , at most two� screws

Fig. 4. One line in the tolerance-zone of Fig. 2 is defined by the coordinates ofM

two points, ( ) within and ( ) within .� �¼ * % ¼ & *T

Z Z

of which (i.e. $ and $ ) can have zero pitch. All the others have non-zero pitch (see�

Ball, 1998 or Hunt, 1990). The purpose of this section is to show that eqn (2) bemay used, to the first order, to identify each line-segment in a tolerance-zone.

When two lines $ and $ are each formed as in eqns (4), and then combined linearly� �

with the relation

$ (5)� � �� � � �$ $ ,� *� �the pitch of $ is found to be a small quantity of the order [ ] , while the coordinates � � �,� ! " �, , and are of the order [ ]. Further, when the quadratic identity is found for $,only terms of [ ] and [ ] appear. After addition, the only remaining terms are of order� �� 4

[ ] . Therefore, to the first order of the tolerance on location, every line in the tolerance-� 4

zone may be represented with combinations provided by eqn (5), and, when usedrepetatively, with combinations provided by eqn (2).

5. The Line-Solid of a Tolerance-Zone

The manufacture of a part with a hole is more expensive when the hole must be locatedaccurately with a smaller tolerance than when it can be located loosely with a largertolerance. Therefore, the cost for manufacture must be related in some inverse fashion tothe magnitude of the line-solid for the tolerance-zone and also to the content ( -�dimensional volume) of the T-map. We need a way to calculate a value both for a line-solid and for the content of all, or a portion of, a T-map.

5.1 Content of the reference simplex in areal coordinates. Given any five points , ,+ ,Z Z

- . /Z Z Z, , and in the space of the reference simplex, each point defined with its arealcoordinates , , , , or relative to , the Grassmann principle for space0 1 2 3Z Z Z Z Z

� � � �� $ $ $ $ $� � � � 5

suggests that the four-dimensional volume (content) of is+ , - . /Z Z Z Z Z

� �� �� �� �� �� �� �� �� �� �

0 0 0 0 01 1 1 1 1 2 2 2 2 23 3 3 3 3

Z Z Z Z Z� � � � Z Z Z Z Z� � � � Z Z Z Z Z� � � � Z Z Z Z Z� � � �Z Z Z Z Z� � � �

(6)

(see e.g. Klein, 1939). This result gives the content as a fraction or a multiple of thecontent for the reference 4-simplex . When forming content as a sum of$ $ $ $ $� � � � 5

several simplexes, it is important in each determinant to maintain the same cyclic order ofpoints so that some terms do not enter the sum negatively.

5.2 The line-solid for a tolerance-zone. A content provided by eqn (6) corresponds to theratio of two line-solids that can each be formulated directly from a tolerance-zone, forinstance the one shown in Figs. 2 and 4. In Fig. 4, the line is located with the four'coordinates , , , and . We can then geometrically construct the four-parameter set� �Z Z � �

of lines in the entire tolerance zone by letting each of these variables, one after the other,assume all values over their ranges. For example, we first could hold , , and fixed� Z � �

and let only range interior of the limits determined by circle ; the result would be a� �Z

partial pencil of lines in a plane parallel to the -plane (see Fig. 2) and with center at the��point . Then when would also be permitted to assume all values over its( � % � & �� � Z

range, the combined result would be the partial bundle of lines which is bounded by acone (shown with dotted lines in Fig. 4) that has its apex at point and that intersects the(� � �Z Z-plane in circle . When, additionally, coordinate is permitted to vary, the result is�

a family of partial bundles, each bounded by a cone that intersects the -plane in circle� �Z Z

� � ���

and that has its apex on the line-segment in circle at a distance from the -plane.�

Then, finally, when coordinate is free, the apexes of the bounding cones occupy the� all points within the circle , and the construction is complete for the line-solid that forms�

�

the tolerance-zone. Of course, a similar construction (that leads to the same result) can

be produced by using polar coordinates , , , and to locate points 4 !Z Z� � � and ,(respectively. Mathematically, the line-solid is obtained with the quadruple integral�

or (7)

d d d d

d d d d

��� � � �� � � �

�

�

)

)

� �

4

Z Z

� � � �

!«� � !«� �Z Z

� �

� � �� �

Therefore, for the tolerance-zone shown in Fig. 4, the line-solid is the product of the�

areas of the two circles and , i.e. .� � � �5�

� �

One important characteristic of the pattern of lines in the tolerance-zone is the linearrelationship between angular orientation and the projection onto the -plane of the line-��segment . Because of the linearity, the operations of integration in either form of(!expressions (7) are commutative and the construction above yields exactly the same line-solid when the four coordinates , , , and are taken in any order.� �Z Z � �

6. The Line-Solid of the Reference 4-Simplex

The line-solid that corresponds to the content (four-dimensional volume) of the reference4-simplex is the solid of those lines only that correspond to points in eqn (2)$ $ $ $ $ � � � 4 5

for which ( ). The pierce-points of , $ , and $ with the planes� 6 6 � � � � ��� � � �$of circles and , respectively, define a lower and an upper isosceles right triangles� �

�

(Fig. 2), each having the leg-dimension . It is helpful to decompose the desired line-���

solid into lines that pierce both isosceles triangles (a zone of lines fitting in a triangularright prism) and additional ones that pierce the planes of circles and inside, or on the� �

�

boundary of, just one of these triangles. Taking first the line-solid for the prism, it is��/64 (see §5.2) because its cross-section is the isosceles right-triangle of leg-dimension��� �

�. Next we determine the line-solid for those lines that pierce the plane of circle at

points that are either in, or on the boundary of, the upper isosceles triangle but pierce theplane of circle outside the lower right triangle. Point is represented as a differential� (element that is located in the plane of circle with the coordinates and ; the element� 7

��

has dimensions d d 2. The hatched area in Fig. 5 represents all those lines through� 7� ( � � 6 6 that pierce the plane of outside the lower right triangle and for which .��

The numerical labelings in Fig. 5 correspond to the lines that are used in linearcombinations (all ) to obtain the given pierce-point. For instance, point takes�� 8 � 9the designation 14 because it is a point in the plane of obtained from a linear�combination of $ and $ ; line 143 represents those points from linear combinations of� �

the line 14 (i.e. through ) and $ , etc. The area of the shaded region in Fig. 5 is 9 � �:� �

for any . Lastly, we note that there is another line-solid for those lines that pierce the�

plane of outside the upper right triangle, thereby doubling the content for those lines��

not in the prism defined by the two isosceles triangles. Therefore, the entire content ofpoints the reference 4-simplex corresponds to the line-solidinside

Fig. 5. True view of circle in Fig. 2 from which . The* +6 y ,6 y* * �

hatching identifies the region that is pierced by lines in the referencehyperhedron and through in Fig. 2 less the isosceles triangle of area 8. The: ! «�

diameter of the circle is ; dimension is the same value as in Fig. 2.!

�� � �� � �� � � 7

� �

5: :

� � � �� � ;

5: :< =�

�

� �

!«�

� ��

2d d

.

(8)

�l2

According to §5.1, using areal coordinates, the content of the entire tolerance-zone forlocation (Fig. 2) is the ratio , i.e. . � � � �� �5 � ;� �=� � �;

7. A Tolerance on Perpendicularity

The Y14.5M standard (1994) specifies a tolerance on for a target axis asperpendicularity a floating cylindrical zone that is exactly perpendicular to a datum plane or other axis andwithin which the target axis (line) must lie. For instance, the tolerance on�ZZ

perpendicularity ( ) applies to the bigger hole in Fig. 1 relative to Datum , the� > � �ZZ

lower surface of the part. The effect in Fig. 6 is to limit the variation on orientation of theaxis to lines that can lie in the smaller cylinder of diameter , but not to further limit�ZZ

variation on location; the smaller cylinder for perpendicularity can translate laterallyanywhere within the tolerance-zone . Consequently, the T-map for the axis of the larger�hole in Fig. 1 is represented with the three sections shown in Fig. 7, i.e. the three sectionsof Fig. 3 in which those of Figs. 3( ) and 3( ) now have a triangle removed (dotted);� �there is no change to the boundary of the T-map for the cross-section in Fig. 3( ). In all�three cross-sections of Fig. 7 the T-map for the lines in the orientational tolerance-zone(diameter ) of Fig. 6 is shown with dashed lines; it can only translate in the - and -� ! "ZZ

directions. The increment in cost for manufacturing the hole must be related to the line-solid that represents those lines that can by the axis, i.e. the line-no longer be occupied

solid obtained as the difference of the line-solid given by (eqn (7)) and the line-� �� �5solid that is swept by the orientational tolerance-zone of diameter .�ZZ

Fig. 6. The tolerance-zone !ZZ on perpendicularity is shown both at the center ofthe locational tolerance-zone and at one position displaced radially d from the! �

center. Element d d is shown with heavy shading.� ��

7.1 Sweeping of a line-solid. A tolerance on perpendicularity for an axis requires that �ZZ

the tolerance-zone for location (tolerance , ) be formed by the sweeping of the� � 8 �ZZ

smaller tolerance-zone. This sweeping can be modeled as a succession of infinitesimallyseparated circumferential sweeps about circles of radius , the innermost of which occurs4about a circle of radius d (Fig. 6). Clearly the swept zone will be a sum of the line-solid4

contained in the cylinder of diameter , i.e. , plus additional lines that are � � �5ZZ � ZZ�

encountered with the circumferential sweeps.Taking first the circumferential sweep about a circle of radius d , half of the new4

lines created are those passing through an element d d in the plane of 4 4� ��

(Fig. 6) andall the points of the area � � �ZZ� 4 lying in the plane of . The other half is for theelement d d that lies in the plane of circle and the area 4 lying in the plane of4 4 � � �� ZZ�

� ��

. Then all the lines in the swept locational tolerance-zone of diameter are

��� �

� � � 4 4

5 :

� !«� �

�

� �

� �� * �

ZZ ZZ� �

! «�

ZZ � ZZ� �

� �� �

ZZ

d d�

�

5� � �.

(9)

When = , then the swept solid of lines becomes ( ) , or 7/16 of the� � � ;���5 �ZZ � ���

�

value when no orientational tolerance is specified. The solid of lines that is removed byspecifying the orientational tolerance is then 9/16 of the locational line-solid in Fig. 2.�ZZ

Fig. 7. The effect on the T-map of Fig. 3 when a tolerance ( ) for! ! z !ZZ ZZ

perpendicularity is also specified. The floating T-map for perpendicularity(dashed lines) can only translate parallel to the - and -axes.7 8

8. The Tolerance-Map (T-Map) for a Cylindrical Surface

A cylindrical surface can be represented with a line (axis) and a diameter or radius tospecify size. The screw is a mathematical entity perfectly suited to specifying a cylinder.One can regard the entire space of screws as all the lines in space for each of which a fifthparameter ranges over the real numbers (Ball, 1998; Hunt, 1990). In the traditional usesfor screws, e.g. in expressing any system of forces as a wrench, the fifth parameter is the

pitch, i.e. the ratio of the couple that acts parallel to the resultant force. For our purposewe will use the fifth parameter to express values that range from zero to the tolerance on the size of a cylindrical surface that is centered on a line. Because the coordinates ofthe zero-pitch screws $ $ in eqn (1) are partially separable, i.e. coordinates and� � � �# do not distinguish between lines, we now add another term $ to eqn (1) in which $�

is a screw of infinite pitch and has Plücker coordinates (0 0 0 0 0 ). The size-� � � � tolerance then resides only in the Plücker -coordinates of $ and $ # � in the equation

� ��� y�

� � � � � � � � � � � � � � � �$� � � � � � �$ $ $ $ $ $ . (10)

Then, by making suitable choices for the coordinates areal ��� ��6, $ represents any�

one of the five-parameter set of that are represented by the tolerancescylindrical surfaces � and for location and size, respectively, used in combination.

9. Conclusion

We have proposed a new mathematical model for the geometric tolerances of lines andcylindrical surfaces which utilizes the geometry of lines and screws. It is based on ahypothetical volume of points which we call a Tolerance-Map. T-maps can be resolvedinto component zones that illustrate the quantitative effect of interdependencies that arestipulated in the ISO/ANSI/ASME Standard (1994). Based on the premise that cost ofmanufacturing is in some way proportional to the content ( -dimensional volume of�lines) of a Tolerance-Map for location of an axis, we present ways to compute the line-solid that describes the totality of possible locations of an axis in a tolerance-zone.Expressions for the line-solid are developed for the location of an axis and for therestrictions that an orientational tolerance provides.

In this paper we treat tolerances on location and orientation; in future papers we planto incorporate other types of tolerances.

Acknowledgements

The authors are grateful for funding provided to this project by National ScienceFoundation Grant #DMI-9821008.

ReferencesAmerican National Standard ASME Y14.5M. (1994). , The AmericanDimensioning and Tolerancing

Society of Mechanical Engineers, NY.Ball, R. S. (1998). . Cambridge University Press. The Theory of Screws [Originally published 1876 and

revised by the author in 1900; now reprinted with an introduction by H. Lipkin and J. Duffy.]Chase K., Gao J., Magelby S., Sorensen C. (1998). Including geometric feature variations in tolerance

analysis of mechanical assemblies, , , pp 795-807.IIE Transactions 28

Clement A., Desrochers A., Rivière A. (1991). Theory and Practice of 3D tolerancing for assembly,Second CIRP Seminar on Comp. aided Tolerancing, PennState, May.

Clement A., Riviere A., Temmerman M., Cotation tridimensionelle des systemes mecaniques, theorie etpratique, in French, Yvry-sur-siene. (English translation in progress by Addison-Wesley, Boston).

Coxeter, H. S. M. (1969). , 2nd ed. Wiley.Introduction to GeometryForder, H. G. (1960). . Chelsea.The Calculus of ExtensionDesrochers, A. (1999). Modeling Three Dimensional Tolerance Zones Using Screw Parameters. CD-

rom Proceedings, ASME Design Automation Conference, Paper #DETC99/DAC-8587, 9 pp.Desrochers, A. and Rivière, A. (1997). A Matrix Approach to the Representation of Tolerance Zones and

Clearances. , , 6630-636.Int'l J. of Advanced Manufacturing Technology 13Ghosal, A. and Roth, B. (1987). Instantaneous Properties of Multi-Degrees-of-Freedom Motions—Line

Trajectories. , , 116-124.ASME J. of Mechanisms, Transmissions, and Automation in Design 109Gossard, D. C., Zuffante, R. P. and Sakurai, H. (1988). Representing dimensions, tolerances, and features

in MCAE systems, . (2):51–59.IEEE Comp. Gr. and Appl 8Hilbert, D. and Cohn-Vossen, S. (1952). . Chelsea. (Trans. from theGeometry and the Imagination

German by P. Nemenyi.)Anschauliche Geometrie Hillyard, R. C., and Braid, I. C. (1978). Analysis of dimensions and tolerances in computer-aided

mechanical design, (3):161–166.Computer-Aided Design, 10Hunt, K. H. (1990). . Clarendon Press, Oxford. [Originally publishedKinematic geometry of mechanisms

1978; now reprinted with corrections.]Hunt, K. H. (1976). Geometry—The Key to Mechanical Movements. ,Mechanism & Machine Theory

11:79-89.Klein, F. (1939). Elementary Mathematics from an Advanced Standpoint, Geometry. MacMillan.

(Trans. from the 3rd German ed. (1925) by E.R. Hedrick and C.A. Noble).Kramer, G.A. (1992). , MIT Press.Solving Geometric Constraint System: A Case Study in KinematicsLight, R. A., and Gossard, D. C. (1982). Modification of geometric models through variational geometry,

Computer-Aided Design, 14(4):209–214.Martinsen, K. (1993). Vectorial tolerancing for all types of surfaces, Proc. of 19th ASME Design

Automation Conf., Albuquerque, Vol. 2, ASME Press.Requicha, A. A. G. (1983). Toward a theory of geometric tolerances, Int. J. of Robotics Research

2(4):45–60.Requicha, A. A. G., and Chan, S. C. (1986). Representation of geometric features, tolerances, and

attributes in solid modelers based on constructive geometry, IEEE J. of Robotics and Automation, RA-2(3):156–166.

Reuleaux, F. (1875). . Vieweg,Theoretische Kinematik: Grundzüge einer Theorie des MaschinenwesensBraunschweig. Trans. Kennedy, A. B. W. (1876) as . MacMillan, London.The Kinematics of MachineryReprinted (1963) with a new introduction be E. S. Ferguson, Dover, NY.

Rivest, L., Fortin, C. and Morel, C. (1994). Tolerancing a solid model with a kinematic formulation,Computer Aided Design, , June.26

Roy, U., and Liu, C. R. (1988). Feature-based representational scheme of a solid modeler for providingdimensioning and tolerancing information, , (314):333–Robotics and Computer-Integrated Manufacturing 4345.

Samper, S. and Giordano, M., (1998). Taking into Account Elastic Displacements in 3D TolerancingModel and Application. , , 156-162.J. of Materials Processing Technology 78

Sommerville, D. M. Y. (1929). . Methuen, London.An introduction to the geometry of dimensionsnTischler, C. and Samuel, A. (1999). Prediction of the slop in general spatial linkages. Int'l J. of Robotics

Research, , 845-58.18Turner, J. U. (1990). Exploiting solid models for tolerance computations. In Geometric Modeling for

Product Engineering, (M. J Wozny, J. U. Turner, and K. Preiss, eds.), pp. 237–258, North-Holland.

Appendix: Polytopes and the 4-Simplex

In three dimensional space, regular polyhedra are formed by joining regular polygonsalong edges and at vertexes; one of these is the regular tetrahedron. Non-regularpolyhedra are formed the same way, but now at least one polygon has unequal angles atits vertexes. In spaces of higher dimension regular and non-regular hypersolids can alsobe formed, and the term for them is . For example, in four-dimensions there arepolytopesix regular polytopes, one of which is the regular 4-simplex (see e.g. Hilbert and Cohn-Vossen, 1952 or Sommerville, 1929). In -space the regular tetrahedron for dimension 3�generalizes to the - . The simplex occurs in spaces of all dimensions: in� simplexdimension zero, it is a point; in dimension 1, a line-segment; in 2, a triangle; in 3, atetrahedron; in 4, a 4-simplex; etc. Non-regular simplexes occur in all spaces ofdimension 2 and higher; they are formed simply by spacing some or all of the vertexes atdifferent distances from each other.