Int. J. Product Lifecycle Management, Vol. x, No. x, xxxx
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Mathematical theory of dimensioning and parameterising product geometry Vijay Srinivasan IBM Corporation and Columbia University, New York, NY, USA E-mail:
[email protected] Abstract: The classic art of dimensioning a drawing and the modern art of parameterising a product geometric model in a computer aided system have a lot in common. As industry moved into the information age, this commonality became apparent and the need to find a unified mathematical theory assumed greater urgency. This paper presents the results of nearly two decades of work towards such a theory. It is based exclusively on congruence theorems, some of which are known since the days of Euclid and others which were proved only recently. These results are getting incorporated into national and international standards that deal with product lifecycle management. Keywords: dimensioning; parameterising; geometry; product; congruence; symmetry; relative positioning; constraints; solids; standards. Reference to this paper should be made as follows: Srinivasan, V. (xxxx) ‘Mathematical theory of dimensioning and parameterising product geometry’, Int. J. Product Lifecycle Management, Vol. x, No. x, pp.xxx–xxx. Biographical notes: Vijay Srinivasan in the Programme Manager for PLM Research, Standards, and Academic Programs at IBM. He is also an Adjunct Professor of Mechanical Engineering at Columbia University in New York City, where he teaches courses in CAD/CAM. He joined the IBM Research Division in 1983 and has been involved in CAD/CAM research since then. Dr. Srinivasan is a member of several national and international standards committees. He is a member of the Technical Advisory Committee of the standards consortium PDES Inc and the Vice-Chairman of the US Technical Advisory Group to ISO/TC 213. He has published widely, and his book Theory of Dimensioning was published by Marcel-Dekker in 2004.
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Introduction
Geometry is an attribute of a product that manifests itself prominently throughout the lifecycle of the product. The overall shape and size of the product are determined during the conceptual design stage. They are refined further by theoretical and experimental analyses, when the geometry is optimised for performance and function of the product. Detailed design identifies critical dimensions and tolerances before serious manufacturing begins. Manufactured parts are inspected for geometric conformance to specifications before they are delivered to customers. Geometry continues to be a matter of concern even when the product is maintained and serviced, till it is retired. Therefore it is not surprising that industry pays enormous attention to product geometry. Copyright © 200x Inderscience Enterprises Ltd.
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Till recently, product geometry was captured exclusively using engineering drawings, a practice that continues even today in some organisations. These drawings contain stylised indications of dimensions, tolerances, surface finish, and other relevant information. With the advent of three dimensional geometric and solid modelling using Computer Aided Design (CAD) systems, we can do more. In particular, the geometric model can be parameterised, which enables easy generation and analysis of many instances of the product design. While the well known technique of dimensioning drawings and the more recent use of parameterised geometric models are very popular in industry, it is only recently that a unified mathematical theory has emerged to put these two on a stronger scientific ground (Srinivasan, 2004). Such a theory has two major benefits: first, it provides a conceptual clarity on which we can build further advances, and second, it lays out a rational basis to develop software systems for product design and manufacture. So what is the new unified theory of dimensioning and parameterising product geometry? It is based on the simple but powerful concept of congruence. Section 2 explains how the notion of congruence can form the basis for dimensioning and parameterising a family of geometric models. This leads to a comparison of old and new taxonomies of dimensioning in Section 3. Intrinsic dimensions form the leaf nodes of the hierarchy in the modern dimensional taxonomy, and these are explained in Section 4. The rest of the hierarchy depends on relational dimensions addressed in Section 5, where symmetry plays an important role. Geometric and dimensional constraints, and dimensioning solids are the topic of Section 6, before a summary and concluding remarks are presented in Section 7.
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Congruence
Let us start with the simple question of how to dimension triangles. And while we are at it, can we also explore how to parameterise them? Figure 1 shows some successful attempts to answer these questions. It seems intuitively obvious that all these three schemes are valid ways of parameterising triangles, and we get valid dimensions when numerical values are assigned to the distances and angles indicated by arrows. But what is the theoretical basis for our intuitive belief that these are valid? A little reflection shows that each of the three schemes in Figure 1 can be associated with a famous triangle congruence theorem from Euclid’s Elements (Heath, 1956): Figure 1(a) with the side-angle-side theorem (Book I, Proposition 4), Figure 1(b) with the angle-side-angle theorem (Book I, Proposition 26), and Figure 1(c) with the side-side-side theorem (Book I, Proposition 8). Figure 1
Examples of dimensioning and parameterising schemes for triangles
(a)
(b)
(c)
Mathematical theory of dimensioning and parameterising product geometry
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This simple exercise with triangles provides several insights. •
Dimensions are merely numerical values assigned to certain geometric parameters; in fact, dimensions are instantiations of distance and angle parameters.
•
Congruence theorems seem to provide theoretical justifications for dimensioning and parameterising schemes; this idea will be explored further.
•
There is more than one way to dimension or parameterise a triangle, but in each case only three dimensions or parameters are needed. In other words, a triangle belongs to a three parameter family of geometric objects.
We also notice some constraints that should be observed in dimensioning. In Figure 1(a) the angle dimension should be less than 180º. In Figure 1(b) the sum of the two angle dimensions should be less than 180º. In Figure 1(c) the sum of any two side dimensions should exceed the third side dimension – this is the familiar triangle inequality constraint. The combination of notions such as n-parameter family of geometric objects, their instantiations in dimensioned objects, and constraints imposed upon these dimensions directly suggest how one may be able to define data models for such entities using schemas and their instances. Such schemas and schema instances form the basis for standardised data models for geometric objects. In addition, they guide the definition of object classes for writing geometric software. The schemes shown in Figure 1 are not the only ones to dimension or parameterise triangles. Figure 2 shows three other schemes that one may try. Are they valid, in the sense that we can produce a unique triangle with the dimensional specification in each case? In these cases, it is instructive to see if we can draw these triangles using ruler, compass, and protractor. It is easy to see that the scheme in Figure 2(a) is valid because it can be based on a simple extension to the angle-side-angle theorem. The scheme of Figure 2(b) is not valid because there is another triangle that has the same dimensions. (Can you sketch this?) It turns out that the scheme of Figure 2(c) is valid because this triangle can be drawn in a finite number of steps using ruler, compass, and protractor. Try to draw this – it is a non-trivial exercise. In fact, we can propose a base-height-vertical angle congruence theorem to support this dimensioning scheme. Figure 2
Further examples of dimensioning and parameterising schemes for triangles
(a)
(b)
(c)
Before we leave the topic of congruence, let us observe that the classic congruence theorems of Euclid actually mix two types of congruence in two dimensional cases. Figure 3 shows these two types: Figure 3(a) shows two triangles that can be superposed on each other by moving one relative to the other without leaving the plane that contains them, while Figure 3(b) shows two triangles that require that we move one out of the plane, by flipping, to superpose it on the other. It may be observed that the two triangles in Figure 3(b) are mirror images of each other in the plane. This distinction may not seem
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to be an issue as long as we deal with two dimensional objects that are embedded in three dimensional space. But, it matters a great deal when we deal with truly three dimensional objects. Figure 3
Illustration to two types of congruence
(a)
(b)
Figure 4 shows two general tetrahedra (that is, nonregular tetrahedra) that are mirror images of each other. These are not congruent under rigid motion in three dimensional space, even though some geometers consider them to be congruent under isometry (that is, under congruent distances and angles.) Such objects are called chiral objects. In fact, the best definition of chirality was given by Lord Kelvin about hundred years ago: “I call any geometric figure, or group of points, chiral, and say it has chirality, if its mirror image, ideally realised, cannot be brought to coincide with itself.”
It is obvious that chiral objects are not geometrically interchangeable for engineering applications. A right handed glove is a mirror image of a left handed glove, but these are not generally interchangeable. Similarly, left and right handed screw threads are not interchangeable, because they are chiral objects. So, for engineering applications, we need to formulate congruence theorems that take chirality into account. Given the apparent importance of chirality at such a basic level, it is surprising that it is not sufficiently emphasised in engineering education. It is interesting to note that chirality is studied with great interest in organic chemistry. The carbon atom has an outer valence of four. So, many organic molecules that involve carbon have tetrahedral arrangement and hence end up being chiral objects. And these chiral molecules can have dramatically different biological properties. This has driven the study of chirality to a high level of prominence in the pharmaceutical and food manufacturing industries. It is time that we accord similar importance to chirality in general engineering. Figure 4
Illustration of chirality among tetrahedra
Mathematical theory of dimensioning and parameterising product geometry
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To summarise, we can make the following engineering statements – rather loose, but sensible as common notions – regarding congruence under rigid motion: •
Congruent objects are functionally interchangeable. Note that this applies only to congruence under rigid motion, and not under isometry. From now on, we consider only congruence under rigid motion.
•
In industrial parlance, congruent objects have the same ‘part number’.
•
Objects that have the same dimensions must be congruent under rigid motion.
At a more rigorous mathematical level, we observe that congruent objects (under rigid motion) belong to an ‘equivalence class’, because the congruence relation is •
Reflexive: that is, A is congruent to A
•
Symmetric: that is, if A is congruent to B, then B is congruent to A
•
Transitive: that is, if A is congruent to B and B is congruent to C, then A is congruent to C.
Establishing such equivalence class is very important for classification purposes. We are very fortunate that congruence serves as an easy and obvious equivalence relation. This allows us to say that two spheres are equivalent if they have the same radius (because, then they would be congruent) and that the sphere belongs to a one parameter family of surfaces. So it is perfectly valid to parameterise a sphere by its radius and to dimension a sphere by specifying a numerical value for its radius, or, equivalently, its diameter.
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Taxonomies of dimensioning
Dimensioning is older than parameterisation, and has a longer history and experience base in industry. Nearly seventy years ago, Carl Svensen proposed a theory of dimensioning (Svensen, 1935). His theory consists of three parts: size dimensions, location dimensions, and a dimensioning procedure. He classified size dimensions into six types, of which five are shown in Figure 5 – the sixth accounts for those that are not covered by these five types. The positive and negative cases refer to solids and voids, respectively. He arrived at his classification by way of vast industrial experience. This empirical classification turns out to be surprisingly good. Figure 6 shows Svensen’s classification of location dimensions. For this, he depended on geometric elements such as centres, axes, and locating surfaces. Again, these have been very useful types of location dimensions, though they are empirical. Finally, he put them all together in a dimensioning procedure consisting of four major steps: •
Divide the object into elementary parts (positive and negative).
•
Dimension each elementary part (size dimension).
•
Determine locating axes and surfaces.
•
Locate the parts (location dimensions).
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Figure 5
Svensen’s classification of size dimensions
Figure 6
Svensen’s classification of location dimensions
This dimensioning procedure has been successfully used in industry and is still taught in engineering drafting courses. While this served us well in an industrial age dominated by manual (more recently, computer aided) drafting, is it sufficient for the industry that has moved into the information age? How complete is the classification of elementary parts into six types? What is the justification for location dimensioning using centres and axes, and how could we determine such locating elements in all geometric objects? In other words, do we have a theory of dimensioning that is worthy of the information age?
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To answer this question, it is instructive to reflect upon Svensen’s theory of dimensioning and the taxonomy contained in it. He was clearly trying to divide and conquer a complex problem by dividing it into simpler (elementary) problems that can be solved more easily, and then putting these solutions together. But he used only two levels of hierarchy. In a modern theory of dimensioning, we adopt a similar ‘divide and conquer’ approach but we will allow multiple levels of hierarchy. And we will define the size and location dimensions more rigorously. So we need not abandon, completely, the older theory of dimensioning on which generations of designers and draftsmen have been trained. We have to recast it more rigorously using mathematical terms and results that are more compatible with modern information systems, which store product geometry information in computer interpretable representations. The taxonomy of such a modern theory of dimensioning is shown in Figure 7. Figure 7
A modern taxonomy of dimensioning
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At the leaf nodes of the modern taxonomy of dimensioning, we see intrinsic dimensions. These take the place of Svensen’s size dimensions. At the intermediate nodes we see relational dimensions. These replace Svensen’s location dimensions. The hierarchy can be built with as many levels as the product demands. Complex products may have more levels of hierarchy than simpler ones. In a top-down view of the taxonomy, we take a complex product and divide it into two parts, each of which is assumed to be fully dimensioned. We then have to dimension only the relative positioning of one with respect to the other using the relational dimensions. Then we recur on each of the two parts and apply similar technique till we come to elementary objects at the leaf nodes of the hierarchy. These elementary objects are dimensioned using intrinsic dimensions. It is also possible to take a bottom-up view of the taxonomy. Here we start with elementary curves and surfaces, and dimension them using intrinsic dimensions. Then we take two such objects and position one relative to the other using relational dimensions, which results in a new rigid object. We can combine such objects and move up the hierarchy till we reach the root node that defines the product. In any case, it is only the intrinsic dimensions and relational dimensions that need to be defined, and these will be described in the following two sections.
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Intrinsic dimensions
Intrinsic dimensions arise when we try to dimension or parameterise elementary curves and surfaces. These intrinsic dimensions are the geometric characteristics that remain invariant when the curve or surface is moved by purely rigid motion. Remember that we want to base our theory of dimensioning on congruence theorems. Let us start with elementary curves. The simplest curve in a plane is a (unbounded) straight line. It does not have any intrinsic dimension because all straight lines are congruent. So we consider conics, which are second degree implicit curves in the plane. A well known result (Struik, 1953; Rutter, 2000) in plane analytic geometry is the conics classification theorem: Any planar curve of second-degree can be moved by purely rigid motion in the plane so that its transformed equation can assume one and only one of the nine canonical forms. Of these nine, only the six shown in Table 1 are real, in the sense that they are the ones that can be realised in a real plane. This immediately suggests the conics congruence theorem: Two conics are congruent if and only if they have the same canonical equation. Intrinsic dimensions for the real conics are the numerical values assigned to the intrinsic parameters shown in the last column of Table 1. We can now justify how an ellipse can be dimensioned as shown in Figure 8, because two ellipses are congruent if they have the same major and minor axes – as established by Table 1 and the conics congruence theorem. It is also clear that we can dimension an elliptic half space (that is, the region enclosed by an ellipse) in the same way because the congruence theorems are the same.
Mathematical theory of dimensioning and parameterising product geometry Figure 8
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Dimensioning an ellipse. The horizontal dimension is the major axis (value of 2a) and the vertical dimension is the minor axis (value of 2b)
After the conics, it is natural to consider cubics and higher order implicit planar curves for dimensioning. However, classification and congruence theorems similar to the ones for conics are not readily available for these higher order curves. Even for the planar cubics, the classification is a mess in spite of gallant efforts by Newton and other great minds that followed him. This has left us with special cases of cubics that go by such esoteric names as folium of Descartes, cissoid of Diocles, and witch of Agnesi. Fortunately, engineers resort to free form curves when it comes to designing cubic and higher order curves for which congruence theorems can be found more easily. n
A free form curve has a parametric representation of the form p(t ) = ∑ i =0 piϕ i (t ), where pi are the control points and φi(t) are the basis functions (Farin, 1993). The parameter t is constrained to lie within some chosen interval. If the control points lie in a plane we get a planar curve, and if the control points do not lie in a plane we get a space curve. For free form curves we have an invariance theorem: A free form curve is intrinsically invariant under rigid motion of its control points if and only if its basis functions partition unity in the interval of interest. This leads to a free form curve congruence theorem (stated somewhat loosely): Two free form curves, which share the same basis functions that partition unity, are congruent if their control polygons are congruent. Table 1
Classification of real conics Type Ellipse Special case: circle
Canonical equation
x
2
a
2
2
+
y
2
b2
2
=1 a ≥ b
Intrinsic parameters a, b
,
x +y =a
2
a = radius
Nondegenerate conics Hyperbola
x2 a2
Parabola Parallel lines Degenerate conics
Intersecting lines
y2 b2
a, b
=1
2
y – 2lx = 0 x2 – a2 = 0
x2 a2
Coincident lines
−
−
x2 = 0
y2 b2
=0 a≥b ,
l a = half of the distance between the parallel lines tan–1(b/a) = half of the angle between the intersecting lines None
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An immediate consequence of this congruence theorem is that free form curves such as the Bézier curves shown in Figure 9 can be dimensioned by dimensioning the associated control polygons. This reduces, for example, the dimensioning of quadratic Bézier curve (a parabolic arc) in Figure 9(a) to that of dimensioning a triangle, for which we saw many valid schemes in an earlier section of the paper. Dimensioning the cubic Bézier curve in Figure 9(b) is the same as dimensioning the associated control quadrilateral, which may self-intersect. Figure 9
Quadratic and cubic Bézier curves
(a)
(b)
What we have seen for Bézier curves can be extended to other spline curves such as NURBS, because their basis functions also partition unity. Note that the free form curves we have discussed also include space curves. Before we leave curves, it is worth pointing out that there is a fundamental theorem from differential geometry (Lipschutz, 1969) on the existence and uniqueness of curves: Let κ(s) and τ(s) be arbitrary continuous functions on a ≤ s ≤ b. Then there exists, except for position in space, one and only one space curve C for which κ(s) is the curvature, τ(s) is the torsion and s is a natural parameter (i.e., arc length) along C. This leads us to the mother of all congruence theorems for curves: Two curves are congruent if and only if they have the same arc-length parameterisation of their curvature and torsion. However, this theorem has proved to be of limited use for the practical task of dimensioning curves. Having looked at the dimensioning of curves in some detail, we can predict how these ideas can be extended to the task of dimensioning and parameterising surfaces. Solid analytic geometry (Olmsted, 1947) provides us with a compact classification of quadrics, which are implicit surfaces of the second degree. This immediately furnishes a congruence theorem for quadrics, and this can be exploited for dimensioning and parameterising quadrics and quadric half spaces. Free form surfaces, such as Bézier surfaces and NURBS surfaces, can be proved to be congruent if their control nets are congruent – so we need only to dimension or parameterise the control nets.
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Relational dimensions
Relational dimensions deal with dimensioning (and parameterising) the relative positions of geometric objects. The theory of relative positioning consists of two parts:
Mathematical theory of dimensioning and parameterising product geometry 11 •
A special theory of relative positioning that involves only points, lines, planes, and helices
•
A general theory of relative positioning that reduces the relative positioning of any set of arbitrary geometric objects to that of relative positioning points, lines, planes, and helices.
The general theory of relative positioning greatly simplifies the task of relational dimensioning because we then need to prove only a limited number of congruence theorems. The simplification depends on some interesting results on symmetry. In fact, symmetry plays a key role in developing relational properties among geometric objects. To gain an intuitive appreciation for the role of symmetry, consider the problem of positioning an arbitrary object, such as a chair, in three dimensional space. It requires six dimensions – three for translation and three for rotation. Now consider positioning a sphere in space. It seems to require only three dimensions, which are needed locate the centre of the sphere. We do not need any dimension to specify rotation because the symmetry of the sphere renders all rotations about its centre irrelevant for positioning purposes. Finally, consider the task of positioning a sphere relative to a (unbounded) plane. A little reflection indicates that we need to specify only one dimension, namely the distance between the centre of the sphere and the plane. This drastic reduction in the number of needed dimensions is due to the fact that the plane also possesses some symmetry because it remains invariant under all translations along the plane and all rotations about any axis perpendicular to the plane. So we are right to suspect that symmetry should play a role in relative positioning of geometric objects. We need a simple mathematical device to link the relative positioning problem to congruence theorems. It is the notion of tuples. A tuple is an ordered collection whose members are symbolically enclosed by parentheses, and it satisfies two important properties: •
Tuple equality: (S1, S2, …, Sn) = (P1, P2, …, Pn) if and only if Si = Pi for all i
•
Tuple rigid motion: r(S1, S2, …, Sn) = (rS1, rS2, …, rSn), where r is a rigid motion.
There is a simple mechanical model for a tuple. Intuitively, a tuple represents a collection of objects rigidly welded together by an invisible welding material. When one member in the tuple is moved, all other members also move together, as though they all belong to a single rigid body. Armed with the tuple notion, we can formulate congruence theorems for the special theory of relative positioning. For example, it is intuitively clear that a point in space can be positioned relative to another point in space by specifying only one dimension, namely the distance between these two points. This intuition can be formalised by the following congruence theorem: Let p1, p2, p′1 , and p′2 be points, in a plane or in space. Then (p1, p2) is congruent to ( p′1 , p′2 ) if and only if d(p1, p2) = d( p′1 , p′2 ). Here d denotes the distance. Though this is a simple example, the reader will begin to appreciate the power of tuple notion when more complex problems are considered. The number of combinations of a two tuple, drawn from points, lines, planes, and helices is rather small. Explicit congruence theorems for them, as seen above for two points, can be proved for all of them, thus completing the special theory of relative positioning. It is worth noting an interesting case that arises when a tuple of skew lines as
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shown in Figure 10 is considered. Such a tuple is chiral, unless the twist angle is 90°. So a congruence theorem in this case takes the following form: Let l1, l2 be two skew lines in space, and l′1 , l′2 be two other skew lines in space. Then (l1, l2) is congruent to ( l′1 , l′2 ) if and only if they have the same chirality, d(l1, l2) = d( l′1 , l′2 ) and θ(l1, l2) = θ( l′1 , l′2 ). Here d is the distance between the skew lines along their common perpendicular and θ is the twist angle. This tells us how to dimension or parameterise the relative position of two skew lines. Note that the chirality can be encoded in the sign of the twist angle. Figure 10 A tuple of skew lines (l1, l2) is chiral. l3 is perpendicular to both the skew lines
We can now move to the general theory of relative positioning. Here we ask a crucial congruence question: Has the relative positioning of two geometric objects changed when each of them is subjected to arbitrarily different rigid motions? Mathematically, given two sets S1 and S2 we are asking whether (S1, S2) is congruent to (r1S1, r2S2) for arbitrary rigid motions r1 and r2. Answering this seemingly general and difficult question is greatly simplified by the following tuple replacement theorem: The answer to the ‘tuple congruence question’ remains unaltered if we replace the point sets by those in the same symmetry class. It is very fortunate that there are only seven classes of continuous symmetry, which are shown in Table 2 along with simple geometric elements that belong to the same symmetry class. For example, a cylinder belongs to a cylindrical class, and so does its axis because both of them remain invariant under a translation along the axis and rotation about the axis. So the axis becomes a simple replacement for the cylinder for relative positioning purpose. This compact classification is the result of some rigorous mathematical analysis of the connected Lie subgroups of the rigid motion group, which was completed in mid-1990s (Srinivasan, 2004, 1999; Clement et al., 1994). So the general theory of relative positioning reduces the number of congruence theorems to a limited number of cases, which are handled in the special theory of relative positioning.
Mathematical theory of dimensioning and parameterising product geometry 13 Table 2
Seven classes of continuous symmetry
Type
Simple replacement
Spherical
Point (centre)
Cylindrical
Line (axis)
Planar
Plane
Helical
Helix
Revolute
Line (axis) and point on line
Prismatic
Plane and line on plane
General
Plane, line and point
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Constraints and solids
The theory of dimensioning and parameterisation presented thus far, follows the hierarchy first outlined in the taxonomy of Figure 7. But there are times when engineers specify some dimensions simultaneously, leaving the problem of ensuring its validity to the computer aided design system. These fall under the general topic of dimensional constraints. Before we get into this, it is worthwhile to consider a different set of constraints that fall under the category of basic geometric constraints. A hierarchy of the basic geometric constraints is shown in Figure 11. They are incidence (including tangency), parallelism, perpendicularity, and chirality. These are inherited from the transformation group hierarchy inherent in rigid motions. Figure 11 Hierarchy of basic geometric constraints
Now moving to dimensional constraints, Figure 12 shows two examples of specifying a set of dimensions simultaneously. Given numerical values, we are interested in deciding whether these dimensions specify a rigid collection of geometric objects. In general, such rigidity problems are too difficult to solve. But in special cases, they have been successfully solved by inducing a hierarchy among the simultaneous constraints. For example, in Figure 12(a) we divide and conquer the problem by completing triangle ABE using the side-angle-side theorem, triangle CDE again using the side-angle-side theorem, and finally triangle BCE using the side-side-side theorem because by now we have the
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sides BE and CE available to us. A more interesting case is presented in Figure 12(b), where, depending on the numerical values assigned, we may have a rigid structure or a flexible mechanism for the hexagon ABCDEF. A general rule in industrial practice is to restrict the application of dimensional constraints to two dimensional sketches as far as possible, and to keep them local. Figure 12 Examples of dimensional constraints
(a)
(b)
It may seem that engineers are concerned mostly about intrinsic and relational dimensions involving various geometric objects. But, ultimately, they are interested in dimensioning solids (Hoffmann, 1989). To illustrate this point, consider the example shown in Figure 13 that involves only two planes P1, P2 and a cylinder C. The two planes are constrained to be parallel to each other, and the cylinder axis is perpendicular to the planes. One of the parameters is the distance h between the two parallel planes, which provides a relational dimension, and the other parameter is the diameter d of the cylinder, which provides an intrinsic dimension. Even though all the dimensions, parameters, and constraints seem to be have been specified, it leaves open the question about what solid these geometric objects represent. There are at least four different solids shown in Figure 14 that have the same surfaces completely specified in Figure 13. A general rule is that dimensions and constraints should be imposed on a solid representation to avoid such ambiguity. The same is true for parameterising solids. Figure 13 Relative positioning two planes and a cylinder
Mathematical theory of dimensioning and parameterising product geometry 15 Figure 14 Multiple solids that have the same surfaces as in Figure 13
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Summary and concluding remarks
This paper presented a brief introduction to a modern mathematical theory of dimensioning and parameterising product geometry, based on a number of presentations and discussions the author has had with leading experts in this area over the past two years. More details can be found in a recently published book (Srinivasan, 2004). It is a unified theory built on congruence theorems, some of which are known since the days of Euclid (ca 350 BC) and some of which that involved symmetry were proved only within the last decade. This may explain why a rigorous theory for such a basic engineering task as dimensioning has to wait this long. Interestingly, the motivation for developing such a theory came from recent national and international efforts to standardise dimensioning and tolerancing, and product data exchange among CAD/CAM systems. The ASME Y14.5.1 standard on mathematical definition of dimensioning and tolerancing principles (ASME Y14.5.1M-1994, 1995) was first published in 1995, but it focused only on tolerancing. The work reported in this paper was undertaken to provide a mathematical foundation for dimensioning as well. In addition, the ISO/TC 213 is embarked on an ambitious and worthwhile task of providing a rigorous, unified mathematical foundation for dimensioning, tolerancing, and related metrological practices (ISO/TS 17450-1, 2003). Some of the work presented in this paper is already incorporated in some of the technical specification documents issued by this international standards committee. Finally, establishing the link between dimensioning and parameterising product geometry, assumed some urgency when ISO/TC 184/SC 4 started preparing a standard for representing parameters and constraints on geometric models (ISO 10303-108, 2004). We believe that the work reported in this paper will also lead to a more rational approach to parametric design in industry.
Acknowledgement and a disclaimer The author wishes to acknowledge numerous colleagues in various national and international standards committees, research colleagues in industrial and academic laboratories, and his students at Columbia University on whom much of this technical material was first tried in a course on geometric modelling. Without their kind help and patience, this work would not have seen the light of day. The author’s opinions expressed in this paper are his own and do not represent the official position of the national and international standards bodies.
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References ASME Y14.5.1M-1994 (1995) Mathematical Definition of Dimensioning and Tolerancing Principles, The American Society of Mechanical Engineers, New York. Clement, A., Riviere, A. and Temmerman, M. (1994) Cotation Tridimensionnelle des Systemes Mechaniques, Ivry-sur-Seine, PYC Edition, France. Farin, G. (1993) Curves and Surfaces for Computer-Aided Geometric Design, 3rd ed., Academic Press, San Diego, CA. Heath, T.L. (1956) Euclid: The Elements, Dover, New York. Hoffmann, C.M. (1989) Geometric and Solid Modeling, Morgan Kaufmann, San Mateo, CA. ISO 10303-108 (2004) Parameterization and Constraints for Explicit Geometric Product Models, International Organization for Standardization, Geneva. ISO/TS 17450-1 (2003) Geometrical Product Specification (GPS) – Model for Geometric Specification and Verification, International Organization for Standardization, Geneva. Lipschutz, M.M. (1969) Differential Geometry, McGraw-Hill, New York. Olmsted, J.M.H. (1947) Solid Analytic Geometry, Appleton-Century-Crofts, New York. Rutter, J.W. (2000) Geometry of Curves, Chapman & Hall/CRC, Boca Raton, Florida. Srinivasan, V. (1999) A geometrical product specification language based on a classification of symmetry groups, Computer-Aided Design, Vol. 31, No. 11, pp.659–668. Srinivasan, V. (2004) Theory of Dimensioning, Marcel-Dekker, New York. Struik, D.J. (1953) Lectures on Analytic and Projective Geometry, Addison-Wesley, Cambridge, MA. Svensen, C.L. (1935) Drafting for Engineers, 2nd ed., Van Nostrand, New York.
