Download Boundary Representation Deformation in Parametric Solid Modeling

Boundary Representation Deformation in
Parametric Solid Modeling
SRINIVAS RAGHOTHAMA and VADIM SHAPIRO
University of Wisconsin—Madison
One of the major unsolved problems in parametric solid modeling is a robust update
(regeneration) of the solid’s boundary representation, given a specified change in the solid’s
parameter values. The fundamental difficulty lies in determining the mapping between
boundary representations for solids in the same parametric family. Several heuristic approaches have been proposed for dealing with this problem, but the formal properties of such
mappings are not well understood. We propose a formal definition for boundary representation
(BR-)deformation for solids in the same parametric family, based on the assumption of
continuity: small changes in solid parameter values should result in small changes in the
solid’s boundary representation, which may include local collapses of cells in the boundary
representation. The necessary conditions that must be satisfied by any BR-deforming mappings between boundary representations are powerful enough to identify invalid updates in
many (but not all) practical situations, and the algorithms to check them are simple. Our
formulation provides a formal criterion for the recently proposed heuristic approaches to
“persistent naming,” and explains the difficulties in devising sufficient tests for BR-deformation encountered in practice. Finally, our methods are also applicable to more general cellular
models of pointsets and should be useful in developing universal standards in parametric
modeling.
Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Non-numerical Algorithms and Problems—geometrical problems and computations; G.2.1
[Discrete Mathematics]: Combinatorics—combinational algorithms; H.5.2 [Information
Interfaces and Presentation]: User Interfaces—interaction styles; I.1.2 [Symbolic and
Algebraic Manipulation]: Algorithms—algebraic algorithms; I.3.5 [Computer Graphics]:
Computational Geometry and Object Modeling—boundary representations; J.6 [Computer
Applications]: Computer-Aided Engineering—computer-aided design
General Terms: Algorithms, Reliability, Standardization, Theory
Additional Key Words and Phrases: Algebraic topology, boundary representation, boundary
deformation, cell complex, parametric editing, persistent naming, solid modeling
This research is supported in part by the National Science Foundation CAREER award
DMI-9502728, DARPA contract N00014-96-G-0168, grant DMI-9522806, and UNIGRAPHICS
Solutions.
Authors’ addresses: Spatial Automation Laboratory, University of Wisconsin—Madison, 1513
University Avenue, Madison, Wisconsin 53706; email: ^[email protected]&,
^[email protected]&.
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ACM Transactions on Graphics, Vol. 17, No. 4, October 1998, Pages 259–286.
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1. MOTIVATION
1.1 Ambiguity of Parametric Solid Modeling
At its very inception solid modeling was synonymous with unambiguous
(informationally complete) representations of homogeneously n-dimensional subsets of the Euclidean space [Hoffman and Rossignac 1996;
Hoffmann 1989; Requicha 1977]. On the other hand, the recent rise of solid
modeling as a principal information medium, first in engineering and now
in consumer applications, probably has to do more with development and
successful marketing of new parametric (“feature-based” and “constraintbased”) user interfaces than with the mathematical soundness of solid
modeling systems. These parametric interfaces allow the user to define and
modify solid models in terms of high-level parametric definitions that are
constructed to have intuitive and appealing meaning to the user and/or
application (see Hoschek and Dankwort [1994] for examples and references). The success of parametric solid modeling came at a high price: the
new solid modeling systems no longer guarantee that the parametric
models are valid or unambiguous, and the results of modeling operations
are not always predictable.
Unpredictable behavior of the new systems is well documented in the
literature,1 and we consider some simple illustrative examples in the next
section. The basic technical problem is that a parametric solid model
corresponds to a class of solids, but there is no formal definition or
standard for this class [Shapiro and Vossler 1995]. As a result, different
modeling systems employ incompatible, ad hoc, and often internally inconsistent semantics for processing parametric models.
The lack of formal semantics manifests itself clearly through the socalled “persistent naming” problem [Hoffmann 1993; Shapiro and Vossler
1995]. In modern systems, every edit of a parametric definition is followed
by a regeneration (boundary evaluation) of the boundary representation
(b-rep) for the resulting solid. Since parametric definitions and constraints
explicitly refer to entities (faces, edges, vertices) in the boundary representation, every such regeneration must also establish the correspondence
between the pre-edit and post-edit boundary representations. The correspondence must persist over all valid edits, hence the term “persistent
naming.” But since the semantics of a parametric family is not well defined,
neither is the required correspondence. All systems seek to establish the
correspondence that is consistent with the user’s intuition, but no reliable
methods for achieving this goal are known today. The problem is illustrated
in Figure 1. Every two instances of parametric definition are related
through a parametric edit of one or more parameters t; each instance has
its own b-rep(t), and the problem of persistent naming is to construct the
map g between the two corresponding b-reps.
1
Chen and Hoffmann [1996], Hoffmann and Rossignac [1996], Hoffmann [1993], and Shapiro
and Vossler [1995].
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Fig. 1.
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The role of persistent naming for a parametric family of solids.
Recently, several schemes for persistent naming have been proposed2
that appear to work well in a variety of situations. But the proposed
techniques are not always consistent among themselves and none come
with guarantees or with clearly stated limitations; they may be able to
alleviate the observed problems in today’s systems, but not solve them—
because the problems have not even been formulated.
1.2 Main Contribution
The diagram in Figure 1 places no constraints on properties of the naming
map g. A commonly cited obstacle to such formal properties stems from the
belief that in the most general case parametric edits do not seem to
preserve the basic structure of the boundary representation. For example,
parametric edits may lead to collapsed and expanded entities, (dis)appearance of holes, and the like. Yet the naming mechanisms proposed in
Capoyleas et al. [1996] and Kripac [1997] clearly rely on the generic
topological structure of the boundary representation as a primary naming
mechanism. Certainly we would expect that the naming problem should
have a unique solution when the parametric changes are small and the
resulting boundary representation does not change very much. Accordingly,
we propose that any notion of a parametric family should be based on the
following principle of continuity.
—Throughout every valid parameter range, small changes in a solid’s
parameter values result in small changes in the solid’s representation.
This statement is made precise in Section 2. We argue that this essential
requirement of continuity eliminates much of the unpredictable and erratic
behavior of modern systems as described in Hoffmann [1993] and Shapiro
and Vossler [1995]. In this article, we use this principle of continuity to
develop a formal definition for boundary representation (BR-)deformation
for solids in the same parametric family. It is important to note that “small
changes” may include limited local collapsing of entities in boundary
representations; in this case, the resulting BR-deformation can actually
change the topology of the boundary representation. A special case of the
topological BR-deformation is a geometric BR-deformation that may modify
the shape of the boundary cells, but the topology and the combinatorial
structure of the b-rep remains unchanged.
Our formulation does not solve the difficult problem of constructing the
naming map g in Figure 1, but it leads to necessary conditions that must be
2
Capoyleas et al. [1996], Kripac [1997], Rappoport [1996], and Rossignac [1996].
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satisfied by any naming map g between boundary representations in the
same parametric family. Loosely, the oriented boundary of each cell in the
pre-edit b-rep must be mapped into the closure of the corresponding cell in
the post-edit b-rep with the same orientation. These conditions are powerful enough to identify invalid updates in many (but not all) practical
situations, and the algorithms to check them are simple. In particular, the
identified conditions give formal justification to the naming mechanisms
proposed in Capoyleas et al. [1996] and Kripac [1997], but also identify
techniques that are not acceptable, because they do not satisfy the assumption of continuity. Finally, our definitions explain the difficulties of devising sufficient conditions for BR-deformation and the associated naming
map g.
A definition for BR-deformation is a prerequisite for extending informational completeness to parametric solid modeling and for creating an
industry-wide standard for data exchange of parametric models. To our
knowledge, there are no competing proposals. A notion of variational class
is prominent in mechanical tolerancing and robustness [Requicha 1984;
Stewart 1992]. Notably, Stewart [1992] proposed a definition of variational
class, which was later used in Andersson et al.’s [1995] study of polyhedral
perturbations that preserve topological form. The assumptions of continuity and continuous deformation are also mentioned informally in the
naming techniques and discussion of Chen and Hoffmann [1996]. Our
definitions are consistent with the earlier definitions, but are stronger and
stated in algebraic topological terms, which are more appropriate for
dealing with boundary representations and the mappings among them.
The proposed notion of parametric family and the principle of continuity
are stated in terms of a particular representation scheme (boundary
representation in our case). But our formulation and results are applicable
to more general cellular representation of pointsets. In Section 5, we
discuss briefly how our approach to parametric modeling may be extended
to deal with arbitrary parametric edits and other representations. Applying
similar principles of continuity to other representations would lead to
different definitions of parametric families, for example, those based on
small changes in CSG representations [Rossignac 1990, 1996; Shapiro and
Vossler 1995]. We do not consider such families in this article because most
extant commercial systems are based on b-reps.
1.3 Examples
The following examples illustrate some of the erratic and inconsistent
behavior that occurs during parametric updates of a solid model. These
examples were created in several versions of two leading commercial solid
modeling systems.3
3
We do not identify the specific systems, because similar technical problems are easily
identified in all parametric modeling systems that rely on b-rep for internal representation of
solids.
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Fig. 2.
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Parametric edits of a simple solid.
The simple solid S shown in Figure 2(a) is sufficient to illustrate many of
the relevant issues. The shown boundary representation was produced from
a parametric definition of S. The parametric definition uses a number of
parameters, including parameters t and d that constrain the locations of
features (slot and hole) with respect to the solid’s edges, as shown in the
figure. The value of t determines the location of edge e 1 , which in turn
determines the location of the hole constrained by parameter d. Consider
what happens during a parametric edit of t. Every time t is changed, the
system attempts to regenerate the boundary representation of the solid and
establish which of the edges should be called e 1 . The result of this
computation is easy to determine by simply observing what happens to the
hole constrained by d, since it explicitly references edge e 1 .
If the change in t is small enough not to produce any changes in the
combinatorial structure of the boundary representation, we would expect to
see no abrupt changes resulting from the edit. In other words, the hole and
the constraint d should move continuously with changes in the value of t.
Instead, Figure 2(b) illustrates an example of situations that have been
observed in earlier versions of some systems. In some cases the hole jumps
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on the face f92 and in other situations the hole surprisingly jumps to the
other side of the edge e92 onto f91 . In either case, the observed behavior
would indicate that e 1 was mapped to e92 ; in the latter case the placement
of the hole was also affected by the orientation of the mapped edge e92
(which is opposite to that of e91 with respect to face f93 ). In both cases the
assumption of continuity is violated, even though the correct continuous
mapping clearly exists. We formally characterize such a mapping in Section
2.3 as geometrically deforming. Similar problems have been identified and
illustrated with “jumping blends” by Hoffmann [Capoyleas et al. 1996]. The
naming techniques proposed in Capoyleas et al. [1996], Chen and Hoffmann [1996], and Kripac [1997], as well as the latest versions of many
commercial systems, are now able to establish the appropriate mapping in
this simple case, but not in many more general situations.
Figure 2(c) shows another parametric update of the same solid, where the
value of t is equal to the radius of the cylindrical slot, and the edge e 1
becomes coincident with e 3 . Few (if any) of the latest commercial systems
can handle such an update in a consistent and uniform fashion. A typical
response in one system is shown in Figure 2(d), which indicates again that
e 1 was mapped to e92 ; another system simply signaled an error indicating
that it could not locate the proper edge (the naming map could not be
found). When the slot was constructed as a “hole,” the latter system
signaled no error and simply mapped e 1 to e92 , with the smaller hole
reappearing on the other side of e92 (outside the solid). Yet another system
might delete the constraint (and the hole) with an appropriate warning
message. Again, intuitively it is clear that continuous change in t should
lead to the collapse of face f 1 and identification of edges e 1 and e 3 to e91 . We
characterize such collapsing maps in Section 2.3 as topologically deforming.
These simple examples clearly show that the naming map is usually
developed in ad hoc fashion, is dependent on the way the solid is created,
and may be internally inconsistent across different features and representation schemes. The matching methods proposed in Capoyleas et al. [1996]
and Kripac [1997] will not always properly handle collapsing maps, although it appears that the techniques in Capoyleas et al. [1996] could be
further adapted.
One may also start with the solid shown in Figure 2(d); that is, create the
hole and constrain it with respect to e92 , and then increase the value of
parameter t. One would then expect the face f93 and the hole to gradually
move to the right; we call the corresponding mapping between pre-edit and
post-edit boundary representations expanding. Unfortunately, a more typical response from a system is the one shown in Figure 2(a), and the result
will certainly differ from version to version and from system to system.
Our final example is shown in Figure 3. Parameter t controls the location
of the cylindrical feature that is joined to the cubical body, and one of the
intersection edges is blended.4 The change in the value of t moves the
4
Blends provide a convenient way to track the names of the edges [Capoyleas et al. 1996].
Throughout the article we ignore the geometric changes in b-rep resulting from blended edges.
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Fig. 3.
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Discontinuous update allowed by the parametric modeling system.
cylinder from one side of the block to the other. The system matches the
blended edge e 1 with the edge e91 in the post-edit b-rep as shown. Is there a
way to determine if this update is correct? Notice that there exists a
continuous map between pre-edit and post-edit boundary representations:
it is a simple rigid body motion that flips S upside-down. But this map is
not consistent with the parametric change in t, which requires that face f 1
is mapped into face f91 . Although some of the proposed naming techniques
will allow such edits, we show in Section 3.4 that no change continuous in t
could produce such a boundary representation.
2. FORMULATION
All boundary representations are cell complexes, and their formal properties, including validity, are rooted in algebraic topology [Hoffmann 1989;
Requicha 1977]. Algebraic topology is also the proper setting for formulating BR-deformation. We use a number of standard concepts (complexes,
chains, continuous maps, etc.) that can be found in many textbooks.5 For
brevity, we only state a few critical definitions and results that are
necessary for formulating and understanding the concept of BR-deformation.
2.1 Boundary Representation as a Cell Complex
The boundary of a three-dimensional solid is an orientable, homogeneously
two-dimensional cell complex. Internally, every b-rep consists of an abstract (unembedded) cell complex that captures all combinatorial relationships between cells (vertices, edges, and faces), and geometric information
that specifies the embedding for every cell in E 3 . We do not deal with
5
Please see Agoston [1976], Armstrong [1983], Kinsey [1993], and Munkres [1984].
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specific data structures in this article, and therefore it is not necessary to
distinguish between the abstract cell complex and its embedding. Thus, we
assume that the boundary of a solid S is represented by an embedded
complex K (the b-rep of S).
Many different cell complexes have been used for boundary representations, including simplicial, polyhedral, and CW (with cells homeomorphic to
disks). Our discussion, definitions, and results are not affected by any
particular choice of a cell complex. As the example in Figure 2 illustrates,
b-reps of many commercial systems may contain more general cells, such as
faces with holes, circular edges, and so on; such cell complexes are also
natural from the user’s point of view. The corresponding formal definition,
which we also adopt in this work, is given by O’Connor and Rossignac
[1990]: a selective geometric complex (SGC) is composed of cells that are
relatively open connected submanifolds of various dimension and are
assembled to satisfy the usual conditions of a cell complex:
(1) the cells in K are all disjoint; and
(2) the relative6 boundary bs of each p-cell s is a finite union of cells in K.
Both SGCs and our definitions are valid in n-dimensions, but for the
purposes of solid modeling, we assume that each p-cell s of K is a
p-dimensional subset of E 3 with 0 # p # 3. The union of all embedded
cells is the underlying space uKu of the complex K. In particular, the set of
points b(S) of every solid S is the underlying space of some boundary
representation K.
The boundary of a homogeneously p-dimensional complex K can be also
characterized algebraically, as an algebraic topological chain ­ C(K). The
precise difference and the significance of this distinction is apparent in
Section 3.
2.2 Cell Maps and Cell Homeomorphisms
The cellular structure of b-reps implies the proper mechanism for defining
the notion of a variational class: we consider the effect of a parametric
change on the b-rep cell by cell. Informally, if K is the pre-edit b-rep(t 0 )
and L is the post-edit b-rep(t 1 ), then the naming map g seeks to establish a
correspondence between cells in K and cells in L. So, for example, when
t 1 5 t 0 , we have K 5 L and it is natural that g should be a cell-by-cell
identity mapping. When the difference t 1 2 t 0 is so small that none of the
cells collapse, the correspondence between the cells does not change, even
though some cells may continuously change their shapes. For example, in
Figure 2(a), small changes in t correspond to a small movement of the hole
on face f 2 and therefore to a change in the shape of f 2 in the post-edit
boundary representation. As the difference t 1 2 t 0 gets larger, so does the
difference between the shapes of the corresponding cells; and eventually
The relative boundary of a cell s is defined as the set difference between the closure of s and
s itself [O’Connor and Rossignac 1990].
6
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some cells may actually continuously collapse into some other cells. This is
exactly the case in Figure 2(c) where face f 1 and all of its bounding edges
collapsed into the edge e91 and its bounding vertices.
In other words, the cells in the post-edit b-rep L are images of continuous
variations of the corresponding cells in the pre-edit b-rep K. This cell-bycell continuous variation is captured by the notion of a continuous cell map
(modified from Kinsey [1993]).
Definition 1 (Cell Map). Let K and L be two b-reps. Map g: uKu 3 uLu is
called a continuous cell map from K to L if:
(1) for each cell s [ K, g( s ) is a cell in L;
(2) dimension( g( s )) # dimension(s);
(3) for every s [ K, g restricted to the closure of s is continuous.
The first two conditions capture the notion of a cell-by-cell map. The third
condition implies that g is a continuous map on every cell that can also be
extended to the continuous map g: uKu 3 uLu on the whole boundary of the
solid [Agoston 1976; Munkres 1984; Postnikov 1983]. By definition, continuity of g requires that g(closure (s)) , closure( g( s )) for every cell s
[Munkres 1975]. But since K and L are finite, and uKu and uLu are closed
and bounded, the stronger condition
g ~ closure ~ s !! 5 closure ~ g ~ s !!
(1)
can be shown to hold [Munkres 1984, p. 215].
In keeping with the common practice in topology literature, a cell map g
is used to denote both a map between two b-reps and a map between their
underlying spaces, depending on the context in which it is used. In the
same spirit, we at times abuse the notation and use s to denote both the
cell or its underlying space, again depending on the context. Thus, cell map
g plays a dual role: it acts as a “naming” map and establishes the
correspondence between the cells of two boundary representations; on the
other hand it guarantees the continuity of variation in the underlying space
(the solid’s boundary).
Before we go any further, it may be instructive to check whether naming
maps can be constructed for the examples of the previous section that are
also cell maps. Consider the naming map g between the two boundary
representations of the solids in Figure 2(a) and 2(c) defined by:
g ~ f 1! 5 e91 ,
g ~ e 3! 5 e91 ,
g ~ e 1! 5 e91 ,
g ~ e 2! 5 e92 ,
and so on; that is, the rest of the cells in Figure 2(a) map to their implied
images in Figure 2(c). It is easy to verify that g is a cell map, because it
satisfies all three conditions of the previous definition: every cell in K is
continuously mapped to a cell in L of the same or lower dimension (note
that face f 1 is mapped to edge e91 ) and the continuity condition is satisfied
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for every cell. In particular, note that the boundary set of the collapsed face
f 1 is mapped into the closure of e 1 .
Now consider the mapping of the same boundary representation in (a) to
the one shown in Figure 2(d):
g ~ f 1! 5 e91 ,
g ~ f 3! 5 f93 ,
g ~ e 3! 5 e91 ,
g ~ e 1! 5 e92 ,
g ~ e 2! 5 e92 ,
and the rest of the cells in (a) map to their appropriate images in (d). It is
easy to check that g is not a cell map; for example, the closure of face f 1
contains the edge e 1 , whereas e92 5 g(e 1 ) is not in the closure of e91 5 g( f 1 ).
Referring to Figure 3, consider the mapping g given by:
g ~ f i! 5 f9i ,
i 5 1, . . . n;
g ~ e j! 5 e9j ,
j 5 1, . . . , m,
where n is the number of faces and m is the number of edges, respectively,
in the boundary representation. This map g trivially satisfies all three
required conditions and is therefore a cell map. However, there is also
another cell map h between the two boundary representations:
h ~ f 1! 5 f92 ,
h ~ f 2! 5 f91 ,
h ~ f 3! 5 f93 ,
h ~ e 1! 5 e91 ,
h ~ e 3! 5 e93 ,
h ~ e 2! 5 e94 .
h ~ e 4! 5 e92 ,
This cell map h maps the loop of edges around face f 1 into the loop of edges
around f 2 and vice versa, and corresponds to a rigid body rotation of 180
degrees about the shown x-axis. Thus, given two boundary representations
K and L there may be more than one valid cell map between them.
For sufficiently small parameter changes, we expect that none of the cells
in the boundary representation collapse and that there is also a one-to-one
correspondence between cells in K and L. In this case, we can strengthen
the definition of a cell map.
Definition 2 (Cell Homeomorphism). A bijective cell map g: uKu 3 uLu is
called a cell homeomorphism between uKu and uLu [Munkres 1984].
Notice that when g is a cell homeomorphism, the second condition in the
definition of the cell map can also be strengthened: the dimension of g( s )
must be equal to that of s for every cell in K. For example, both cell maps g
and h, constructed previously for the transformation in Figure 3, are cell
homeomorphisms.
2.3 Topological and Geometric Deformations
In order for boundary representation L to be in the parametric family of
boundary representation K, there must exist a (not necessarily unique) cell
map g from K to L. Furthermore, the principle of continuity postulated in
Section 1.2 implies that when two boundary representations are in the
same parametric range, one must be continuously deformable into the
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Continuous deformation from K to L.
other, as illustrated in Figure 4. Unfortunately, the existence of a cell map
from K to L alone is not sufficient to ensure that K is deformable into L.
For example, even though the two boundary representations in Figure 3
are related by a cell homeomorphism g, K cannot be deformed into L
without leaving E 3 , because L is a reflection of K. We show in Section 4
that such cell maps can be characterized and detected in a straightforward
fashion. A more difficult situation is illustrated in Figure 5. The boundary
representation of two linked tori is related to the boundary representation
of the two unlinked tori by a cell homeomorphism; yet it is not possible to
deform one into another without leaving E 3 , because the continuity principle is violated for some intermediate values of the shown parameter t. If K
is a b-rep(t 0 ) and L is a b-rep(t 1 ), then the continuity principle requires
that, as t changes from t 0 to t 1 , K continuously deforms into L. These
observations motivate the following definition.
Definition 3 (BR-Deformation). Let K, L be two boundary representations. A map F: E 3 3 I 3 E 3 is called a Boundary Representation
deformation or BR-deformation of K to L if:
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S. Raghothama and V. Shapiro
K is homeomorphic to L, but K cannot be deformed into L without leaving E 3 .
(1) F(uKu, 0) 5 uKu;
(2) F(uKu, 1) 5 uLu;
(3) F(uKu, t) is a cell map that is also continuous in t [ [0, 1].
Without loss of generality, we use symbol I in the preceding definition to
denote the closed interval [0, 1], which corresponds to the normalized range
of valid values for the parameter t. In standard algebraic topological terms,
BR-deformation F is a homotopy [Agoston 1976; Kinsey 1993] and can be
also viewed as a continuous family of cell maps F t , for every value t [ I. In
particular, F 0 is the identity cell map and F 1 is the cell (naming) map g
from K to L.
BR-deformation captures the spirit of the postulated requirement of
continuity and appears to be the weakest possible condition. The obvious
BR-deformation can be found by examination in Figure 4 for the parametric
updates shown in Figure 2. Let us consider again the typical situation
illustrated in Figure 1 with pre-edit b-rep(t 0 ) denoted by K and post-edit
b-rep(t 1 ) denoted by L. Only the map g between the two boundary
representations is known, and the problem is to determine whether an
appropriate BR-deformation F exists. If BR-deformation does exist, then we
say that g is a BR-deforming map.
Definition 4 (BR-Deforming Map). A map g from boundary representation K to boundary representation L is called BR-deforming if F(uKu, 1) 5
g(uKu) 5 uLu for some BR-deformation F.
If one accepts the principle of continuity and the implied definitions
previously proposed, then it may be reasonable to require that a naming
map g for solids in the same parametric family must be BR-deforming. We
constructed the appropriate BR-deformation in Figure 4 by examination,
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and it is easy to see that no BR-deformation exists for the example in
Figure 5. But our definition of BR-deformation involves conditions not only
on K and L but also on all the b-reps in the parametric range [t 0 , t 1 ], which
are typically not known. Furthermore, such a BR-deformation need not be
(and usually is not) unique. Is there a general method for checking if a
given map g is BR-deforming? For example, it may not be clear why no
appropriate BR-deformation exists for the parametric update shown in
Figure 3. We delay discussing the sufficient conditions for existence of
BR-deformation until Section 5. In the next section, we discuss necessary
conditions for the existence of BR-deformation and show through examples
how they can be used in practice to distinguish b-reps that belong to the
same parametric family from b-reps that do not.
Observe that boundary representations in Figure 4(a), (b), and (c) are
related by a continuous family F t of cell homeomorphisms. This corresponds to a special but common and important case of geometric BRdeformation F. The combinatorial structure of the boundary representation
K, including the dimension of all cells, is preserved under geometric
BR-deformation. On the other hand, the further deformation of the solid in
Figure 4(d) is not geometric BR-deformation, because one of the faces is
continuously collapsed into the adjacent edge and the combinatorial structure of the resulting b-rep has changed. It should be clear that geometric
BR-deformation is a special case of the more general topological BRdeformation. We show in the next section that the necessary conditions for
geometric BR-deformation are simpler and are easier to test.
3. NECESSARY CONDITIONS FOR BR-DEFORMATION
3.1 Combinatorial Closure and Star Conditions
The third condition in Definition 1 of cell map g requires that g restricted
to the closure of every cell is continuous. We already used this condition to
test whether the maps for the examples of Section 1 are cell maps. The
closure of a cell s in a complex K consists of s itself and all lowerdimensional cells incident on s. Thus, the closure of a two-dimensional face
includes its bounding edges (which are often represented by intersection of
surfaces), and the closure of a vertex is the vertex itself.
To facilitate development of more convenient algorithmic tests, we now
restate this condition in alternative combinatorial terms, using the Cauchy
definition of continuity in terms of open neighborhoods [Aleksandrov 1956].
The combinatorial equivalent of a cell’s open neighborhood is the star of the
cell. The following definition is modified from Cairns [1968] and Munkres
[1984] and is also consistent with definitions in O’Connor and Rossignac
[1990].
Definition 5 (Star of a Cell). The star of a cell s in a complex K, denoted
St( s ), is the union of s and all cells in K that contain s in their boundary.
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In other words, the star of a cell is the union of the cell with its
neighboring higher-dimensional cells. For example, the star of the edge e 1
of the solid shown in Figure 2(a) is the union of cells {e 1 , f 1 , f 3 }. The star of
a vertex is a union of the vertex with all its adjacent faces and edges. The
star of a face is just the face itself, as there are no three-dimensional cells
in boundary representations.
We can now restate the requirement that g restricted to the closure of s
is continuous, originally expressed by the condition (1), to mean that every
neighboring cell of s is mapped to a neighboring cell of the image cell g( s )
or to g( s ) itself. Remembering that some cells s may collapse into lowerdimensional cells g( s ), this requirement translates into a simple condition
that must be satisfied by every cell [Aleksandrov 1956; Munkres 1984]:
g ~ St ~ s !! # St ~ g ~ s !! .
(2)
The strict inclusion applies when the cell s is mapped to a lower-dimensional cell, and condition (2) becomes an equality when the dimension of s
remains the same under the map g. In the latter case, this condition simply
means that adjacency of all cells is also preserved under the map g.
Let us consider the examples of the previous section to see how the star
condition can be verified. The naming map between solids in Figures 2(a)
and (c) takes the star of e 1 into the union of their respective image cells {e91 ,
f93 }. The star of e91 is the union of cells {e91 , f93 } and the other (hidden)
adjacent face of e91 . This clearly shows that g(St(e 1 )) , St( g(e 1 )). The star
condition (2) is easily verified to hold for other cells as well.
Now consider the mapping of the same boundary representation in
Figure 2(a) to the one shown in Figure 2(d). We already concluded earlier
that this map is not a cell map because the closure condition is violated,
and this can be verified easily by the star condition. The star of e 1 is the
union of cells {e 1 , f 1 , f 3 }, which are mapped to cells {e91 , e92 , f93 }. Since the
star of g(e 1 ) is the union of {e92 , f92 , f93 }, it is clear that g(St(e 1 )) Ü
St( g(e 1 )), and g cannot be a cell map.
3.2 Oriented Cell Maps
As we have seen previously, the requirement on g to be a cell map is a
necessary but very weak condition for BR-deformation. The difficulty with
the example in Figure 3 is that the proposed cell map from K to L does not
preserve the relative orientation between certain cells. A simpler twodimensional example is shown in Figure 6. A cell map defined by g( s i ) 5
s 9i takes the cell complex (triangle) K into its reflected copy L; however it is
not possible to continuously deform K into L without leaving E 2 .
In order to take advantage of the orientation information, we need to
slightly modify the definitions of a cell complex and a cell map by requiring
that all cells are oriented, which is usually the case in most boundary
representations. Orientation of a cell s [ K can be visualized as a (positive
or negative) sense of direction. More specifically, every 0-cell (vertex) can
be oriented positively or negatively, orientation of every 1-cell (edge)
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Cell map between K and L does not preserve orientation.
bounded by two vertices s and t is determined by the order of s and t, and
2-cells (faces) can be oriented either counterclockwise (positive) or clockwise (negative). Cyclic edges that do not have vertices can be also directed
in one of two ways. Oriented cell complexes are shown in Figures 3 and 6.
We can now modify the definition of the cell map to account for orientation.
Definition 6 (Oriented Cell Map). A map g: uKu 3 uLu is an oriented cell
map if it is a cell map and for every oriented cell s [ K, g( s ) is an oriented
cell in L.
It can be shown that orientation is a homotopy invariant and is automatically preserved through continuity [Armstrong 1983], which means that
every BR-deforming map is orientation preserving. In other words, the
sense of direction must be preserved at every point of the solid’s boundary,
and we need to ensure this condition in a cell-by-cell fashion.
Since boundary representation of every solid is orientable [Hoffmann
1989; Requicha 1977], the preceding condition is easily enforced on 2-cells
by requiring that all faces are oriented positively (counterclockwise); we
also postulate that all 0-cells (vertices) have positive orientation. It is more
difficult to state what it means to preserve the orientation on 1-cells; note
however that orientation of a face induces orientation in its bounding
edges, and an edge orientation induces orientation in its bounding vertices.
For example, in Figure 6(a) the assumed orientation of edge e 2 is consistent
with the orientation of e 2 induced by the face f. By contrast, in Figure 6(b)
the orientation induced in e92 by f is opposite to the assumed orientation of
e92 . A similar situation can be observed in Figure 3 for the edges bounding
face f 3 . Additional discussion and examples of assumed and induced
orientation can be found in most texts on algebraic topology, for example, in
Agoston [1976].
3.3 Induced Orientation Condition
Informally, orientation of a cell complex K is preserved under a cell map g
if g maps the oriented boundary of every noncollapsed cell s [ K into the
oriented boundary of the image cell g( s ) [ L. The sole purpose of this
section is to express this statement precisely and in a computationally
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S. Raghothama and V. Shapiro
Top view of oriented boundary representation of the solid in Figure 2.
convenient form, using algebraic topological chains. Given a cell complex K,
a p-dimensional chain, or simply p-chain, is a formal expression
a 1s 1 1 a 2s 2 1 · · · 1 a ns n ,
where s i are p-dimensional cells of K and a i are integer coefficients. Two
p-chains on the same cell complex can be added together by collecting and
adding coefficients on the same cells. The collection of all p-chains on K is a
group denoted by C p (K) for p 5 0, 1, 2, 3. Using chains we can replace
incidence, adjacency, and orientation computations with a simple algebra.
In particular, we define the (algebraic) oriented boundary operation in
terms of chains using only three coefficients from the set {21, 0, 11}. (Such
chains are often called elementary chains [Agoston 1976; Hocking and
Young 1961].)
Definition 7 (Boundary of a Cell). The boundary of a p-cell s is the ( p 2
1)-chain consisting of all ( p 2 1) cells that are faces of s with coefficient
11 if the orientation of s is consistent with the orientation of the face and
21 otherwise.7
The coefficients are the simple algebraic means to compare the assumed
orientation of a cell s with the orientation induced in s by an adjacent
higher-dimensional cell (in the star of s). For example, the oriented
boundary of the 1-cell e 1 in Figure 6 is a 0-chain: ­e 1 5 v 2 2 v 1 , which
implies that edge e 1 starts at vertex v 1 and ends at vertex v 2 . The
boundary of an oriented 2-cell f in the same figure is a 1-chain: ­f 5 e 1 1
e 2 2 e 3 , which tells us that the directions of edges e 1 and e 2 are consistent
with the counterclockwise direction of the face f, whereas the direction of
the edge e 3 is not. The face f 2 in Figure 7 has a hole and its boundary is
also a 1-chain: ­f 2 5 e 6 1 e 7 1 e 8 2 e 2 1 e 9 . By convention, the boundary
of a 0-cell is defined to be 0.
The definition of a boundary for an individual cell s extends linearly to
Here we rely on standard terminology in algebraic topology: “face of s” refers to any
lower-dimensional cell incident on cell s.
7
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the boundary of any p-chain; that is, if C 5 a 1 s 1 1 a 2 s 2 1 . . . 1 a n s n is
a p-chain, then its boundary is a ( p 2 1)-chain,
­ ~ C ! 5 a 1­ ~ s 1! 1 a 2­ ~ s 2! 1 · · · 1 a n­ ~ s n! ,
with the usual rules of chain addition. In other words, the boundary
operator is a linear function ­: C p (K) 3 C p21 (K).
The preceding definitions give precise characterization to the informal
concept of “oriented boundary of every cell” as a chain. But the naming map
g is a cell map and technically cannot be applied to chains; to see what g
does to boundaries of cells, we need to extend g to maps on chains.
Intuitively, a chain map g p takes individual p-cells (which are elementary p-chains) in K into individual p-cells in L, just as the cell map g does
when the dimension p of the cells stays the same. But g may also collapse
some p-cell s [ K onto a lower-dimensional cell in L; since these
lower-dimensional cells do not belong to any p-chains on L, the chain map
is instructed to simply ignore them by setting g p ( s ) 5 0. And once again,
we require that the action of a chain map on individual cells must extend
linearly to arbitrary p-chains.8 Together these three conditions give the
usual definition of a chain map [Kinsey 1993].
Definition 8 (Chain Map). Let g: K 3 L be a orientation preserving cell
map. For each dimension p, define a chain map g p : C p (K) 3 C p (L) as
follows.
(1) If s is a p-cell in K and g( s ) is a p-cell in L, g p ( s ) 5 g( s );
(2) if dim( g( s )) , dim(s), g p ( s ) 5 0;
(3) g p (a 1 s 1 1 a 2 s 2 1 . . . 1 a n s n ) 5 a 1 g p ( s 1 ) 1 a 2 g p ( s 2 ) 1 . . . 1
a n g p ( s n ).
Thus, any BR-deforming naming cell map g induces a family of chain
maps g p in every dimension p 5 0, 1, 2. It is also known [Agoston 1976;
Armstrong 1983; Kinsey 1993] that every such family of induced chain
maps satisfies the following commutative diagram,
gp
C p~ K !
O
¡
C p~ L !
2­
2­
C p21~ K !
gp21
O
¡
C p21~ L ! .
Since an oriented cell s can be viewed as a chain with 0 coefficients
attached to all other cells in the cell complex, the preceding commutative
diagram can be enforced in a cell-by-cell fashion. Specifically, given an
orientation-preserving cell map g, the following simple condition is satis-
8
In other words, chain map g p is a homomorphism [Kinsey 1993].
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fied for each oriented p-cell s [ K.
g p21~ ­ ~ s !! 5 ­ ~ g p~ s !! .
(3)
We refer to this condition as the orientation condition, which is a precise
restatement of the requirement that g map the oriented boundary of every
noncollapsed cell s [ K into the oriented boundary of the image cell g( s ) [
L.
3.4 Examples Revisited
Consider once again the examples from Section 1. Technically, every
naming map g consists of the vertex, edge, and face maps. For brevity, we
only specify the action of g on those cells that are necessary for presentation purposes. For example, in the following we assume that vertices are
mapped to their implied images with a positive orientation and omit
explicit description of the vertex map, even though the (chain of the)
oriented boundary of every mapped edge is determined by the vertex map.
Consider the naming map between the solids in Figures 2(a) and (c). The
cells in the top view of the two solids are shown in Figure 7. The semantics
of the update collapses the face f 1 and its boundary into the single edge e91
and its vertices. This deformation is reflected in the topological BRdeforming naming map g as follows.
g ~ f 1! 5 e91 ,
g ~ e 3! 5 2e91 ,
g ~ e 1! 5 e91 ,
g ~ e 2! 5 e92 .
Notice that g is many-to-one and can be characterized as a collapsing map.
We already know from Section 2.2 that g satisfies the star condition (2).
Let’s check if it also satisfies the necessary condition (3) in terms of induced
chains maps g 1 and g 2 :
g 2~ f 1! 5 0,
g 1~ e 2! 5 e92 ,
g 1~ e 5! 5 0,
g 2~ f 2! 5 f92 ,
g 1~ e 6! 5 e93 ,
g 1~ e 1! 5 e91 ,
g 2~ f 3! 5 f93 ;
g 1~ e 7! 5 e94 ,
g 1~ e 8! 5 e95 ,
g 1~ e 4! 5 0,
g 1~ e 3! 5 2e91 .
A simple check shows that the necessary condition (3) ­( g 2 ( f 1 )) 5
g 1 (­( f 1 )) holds, because
­ ~ g 2~ f 1!! 5 ­ ~ 0 ! 5 0,
g 1~ ­ ~ f 1!! 5 g 1~ e 5 1 e 1 1 e 4 1 e 3! 5 0 1 e91 1 0 2 e91 5 0.
The rest of the cells remain topologically invariant under the naming map,
and it is easy to check that the necessary condition (3) holds for all other
cells in the boundary representation.
Observe that the inverse map g 21 from the solid in Figure 2(c) to (a) is
not BR-deforming; in fact it is not a valid cell map because it maps a
one-dimensional edge e91 to a two-dimensional face f 1 . Nevertheless, such
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maps appear useful and are allowed in many systems. Therefore we say
that a one-to-many relation g is a expanding map if its inverse is a
collapsing BR-deforming map.
Now consider the apparent naming map between the solids in Figures
2(a) and (d):
g ~ f 1! 5 e91 ,
g ~ f 3! 5 f93 ,
g ~ e 3! 5 2e91 ,
g ~ e 1! 5 e92 ,
g ~ e 2! 5 e92 .
We already know that this map g is not BR-deforming (because it is not a
cell map), and it is easy to verify that the necessary condition (3) does not
hold either. The induced chain maps are:
g 2~ f 1! 5 0,
g 1~ e 4! 5 0,
g 2~ f 2! 5 f92 ,
g 1~ e 5! 5 0,
g 2~ f 3! 5 f93 ,
g 1~ e 1! 5 e92 ,
g 1~ e 3! 5 2e91 ,
g1~e2! 5 e92 ,
and the left- and the right-hand sides of Equation (3) evaluate to different
chains:
­ ~ g 2~ f 1!! 5 ­ ~ 0 ! 5 0,
g 1~ ­ ~ f 1!! 5 g 1~ e 5 1 e 1 1 e 4 1 e 3! 5 0 1 e92 1 0 2 e91 5 e92 2 e91 .
Clearly, g is not a collapsing map and the inverse map g 21 cannot be an
expanding map.
Next consider the example in Figure 3. We described two distinct naming
cell maps g and h in Section 2.2 that satisfy the star condition (2). We also
argued that neither of them is appropriate— but for different reasons. We
claimed that the naming map g computed by a commercial system is not
BR-deforming because it is not orientation preserving. Indeed, g k ( s ) 5
g( s ) for all cells in the boundary representation, because no cells have been
collapsed, and in fact there is a one-to-one correspondence between the
pre-edit and post-edit boundary representations. Yet, checking the necessary condition (3) for face f 3 , we see that
­ ~ g 2~ f 3!! 5 ­ ~ f93! 5 2e94 2 e93 2 e92 2 e91 Þ g 1~ ­ ~ f 3!! 5 g 1~ e 1 1 e 2 1 e 3 1 e 4!
5 e91 1 e92 1 e93 1 e94 .
Thus, ­( g 2 ( f 3 )) 5 2g 1 (­( f 3 )) and the same relation can be verified for all
other faces in the boundary representation; this shows that g is orientation
reversing and is not BR-deforming.
But now consider the necessary condition (3) for the second naming map
h for the same solids:
h ~ f 1! 5 f92 ,
h ~ f 2! 5 f91 ,
h ~ f 3! 5 f93 ,
h ~ e 1! 5 2e91 ,
h ~ e 3! 5 2e93 ,
h ~ e 2! 5 2e94 ,
h ~ e 4! 5 2e92 ,
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and so on. Notice that h reverses the assumed orientation of all edges
connecting f 1 and f 2 . Checking the necessary condition (3) on this oriented
cell map:
­ ~ h 2~ f 3!! 5 ­ ~ f93! 5 2e94 2 e93 2 e92 2 e91 5 h 1~ ­ ~ f 3!! 5 h 1~ e 4 1 e 1 1 e 2 1 e 3!
5 2e92 2 e91 2 e94 2 e93 .
Similarly, one can check that map h does satisfy the necessary condition (3)
for all other cells in the boundary representation and is a BR-deforming
map associated with the 180-degree rotation of the solid about the X-axis.
Furthermore, h is a cell homeomorphism and is geometrically BR-deforming because it preserves the combinatorial structure of the boundary
representation, even though it may alter the geometry of some cells.
Unfortunately, this particular BR-deformation is not consistent with the
intended semantics of the parametric edit. In fact in this example, due to
symmetry, there are many other geometric BR-deforming maps (e.g., rotation about the Z-axis by 90 degrees). The guarantee of BR-deformation
should not be confused with the guarantee of the correct semantics of the
parametric update.
4. PERSISTENT NAMING
4.1 Sufficient Conditions for an Orientation-Preserving Cell Map
In the typical scenario described in Section 1, a pre-edit b-rep K and a
post-edit b-rep L are given, and the system attempts to match the cells in K
with the cells in L, that is, to construct the naming map g: K 3 L. The
matching process may or may not succeed. In terms of our definitions, there
are at least three reasons why matching may fail:
(1) a BR-deforming map g may not exist, because L does not belong to the
same parametric family defined by K;
(2) a BR-deforming map g does exist, but the system is not able to find it;
and
(3) a BR-deforming map is not unique, and the system is not able to
determine which map captures the semantics of the edit.9
Whatever the outcome of the matching may be, the correctness of the
result depends critically on the system’s ability to determine if a constructed naming map is in fact BR-deforming. The formal machinery
previously developed is sufficient to determine if a given map g is an
orientation-preserving cell map, which is in turn a necessary condition for
g to be BR-deforming. As our examples have illustrated, these conditions
are powerful enough to detect many (but not all) invalid edits in parametric
9
In some cases, this is in fact the correct answer, because the semantics of some edits has not
been defined unambiguously.
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solid modeling. To summarize, a map g: K 3 L is an orientationpreserving cell map if and only if:
(1) K and L are oriented consistently;
(2) g is an oriented cell map;
(3) if dim( g( s )) 5 dim(s), then g maps the oriented boundary of cell s [ K
into the oriented boundary of the image cell g( s ) [ L.
The first requirement is enforced by assuming positive orientation for all
vertices and faces (oriented counterclockwise). The second requirement
reduces to either the combinatorial closure condition (1) or the star condition (2). The third requirement is expressed by the orientation condition (3)
which enforces the relative orientation between edges, vertices, and faces.
All required conditions can be checked easily for a given map g, but it is
important to understand the applicability and the limitations of the proposed conditions.
Examples of Figures 3 and 6 may suggest that the orientation condition
(3) is unnecessarily complicated because it detects only “global” orientation
reversals as shown previously. But imagine the solid in Figure 3 to be
attached to a planar face of another solid (base), and now we can easily
construct a local orientation reversing map that reflects a portion of the
solid’s boundary. Our experience suggests that such local reversals in
orientation occur frequently in commercial systems, apparently due to
partial and incremental updates in boundary representations. Enforcing
the orientation condition for all cells would eliminate this type of invalid
edit.
The orientation condition (3) may appear to imply the star condition (2),
since the closure of every cell includes its boundary points. To see why this
is false, let us assume that g is an arbitrary (not necessarily cell) map that
satisfies the first two conditions of the cell map (Definition 1). We can still
use g to induce chain maps g p and use the boundary operator as before.
Suppose we determine that the orientation condition (3) holds. Does this
mean that g is an orientation-preserving cell map? More specifically, does
this mean that the star condition (2), or equivalently the combinatorial
closure condition (1), is satisfied as well? Not necessarily. Consider the case
when g collapses a p-cell s [ K to a lower-dimensional cell g( s ) [ L. By
definition of a chain map, g p ( s ) 5 0 and hence ­( g p ( s )) 5 0, irrespectively
of what g( s ) may be. Therefore, it is easy to construct a map g such that
condition (3) is satisfied and yet g is not a cell map. Thus, it should be clear
that the star condition and the orientation condition are not redundant for
BR-deforming maps.
On the other hand, suppose that dim( g( s )) 5 dim(s) for all cells in K.
Since g preserves the dimension of every cell s [ K, the orientation
condition (3) implies that 1-cells bounding every 2-cell s2 are in one-to-one
correspondence with 1-cells bounding g( s 2 ), and 0-cells bounding every
1-cell s1 are in one-to-one correspondence with 0-cells bounding g( s 1 ). In
other words, g takes the closure of every cell in K to the closure of the
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corresponding cell in L, which implies the closure condition in the definition of a cell map and the equivalent star condition (2).
Furthermore, recall that when none of the cells are collapsed, the
(geometrically) BR-deforming map g is a cell homeomorphism. It follows
that the star condition (2) is redundant for geometrically BR-deforming
maps. Thus, to check if a map is geometrically BR-deforming, it is only
necessary that none of the cells are mapped to the cells of lower dimension
and that the orientation condition (3) is satisfied for every cell.
4.2 Comparison with Previously Proposed Naming Methods
Several heuristic methods have been proposed for constructing the naming
maps g 10 in a variety of situations, including those where BR-deforming
maps may not exist. A thorough analysis of all methods would not be
practical in this article, but it is instructive to check if the proposed
techniques are consistent with our definitions when they do apply.
The naming techniques proposed by Kripac [1997] and Hoffmann et al.
[Capoyleas et al. 1996; Chen and Hoffmann 1996] rely on several common
methods for naming cells in boundary representations. In particular, both
start with the names of the primitive surfaces appearing in the parametric
definition and construct the names for individual cells using:
—the list of adjacent cells in the boundary representation; and
—the relative orientation of the cell with respect to the adjacent cells.
In general, these two techniques are not sufficient to uniquely match cells
in the two b-reps, and a number of additional techniques must be employed. For example, Kripac [1997] also uses the history of editing to
distinguish between cells, and Capoyleas et al. [1996] use extended adjacency information and directional information associated with the particular solid construction method. The two naming techniques used by both
Kripac [1997] and Capoyleas et al. [1996] intuitively correspond, respectively, to the two types of conditions that ensure an orientation-preserving
cell map. The use of adjacent cells is a means for extending the continuity
of the naming map to the whole boundary of the solid, which relies on the
combinatorial closure condition (1) and/or star condition (2). The matching
of relative orientation of cells is basically equivalent to our orientation
condition (3).
On closer examination, it appears that Kripac [1997] may map a cell s [
K to g( s ) [ L when all cells adjacent to s are mapped into cells adjacent to
g( s ), without considering the orientation information. But as is clearly
shown by example in Figure 3, such a naming map does not have to
preserve orientation and may not be BR-deforming. By contrast, Chen and
Hoffmann [1996] match s to g( s ) only if the adjacent cells also preserve
orientation. This method of naming will never result in an orientation-
10
Please see Capoyleas et al. [1996], Chen and Hoffmann [1996], Kripac [1997], Rappoport
[1996], and Rossignac [1996].
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reversing naming map. Both methods will find the geometrically BRdeforming map, if it exists and every cell has a unique name. It is not clear
whether geometric BR-deformation is always enforced when heuristic
methods are used to resolve multiple matches between cells with the same
names.
Neither of the preceding methods are able to handle collapsing maps in
general as defined in this article, because both Kripac [1997] and Chen and
Hoffmann [1996] require exact match of the adjacent cells, which in turn
corresponds to the special case of exact equality in the star condition (2).
For example, edge e 1 in Figure 2(a) is adjacent to faces f 1 and f 3 , whereas
face f 1 is no longer present in the post-edit boundary representation of
Figure 2(c). Thus, the proposed naming methods will never match e 1 to e91 ,
as they do not allow the strict inclusion relation implied by condition (2).
More generally, all of the proposed methods assume that the dimension of
the mapped cells will always be preserved, which is not the case for
collapsing and expanding maps. Based on our results, it seems to make
sense to redefine the adjacency conditions to correspond to the star condition (2).
Rossignac [1996] observed that when every cell in a b-rep can be
represented by a Boolean (set-theoretic) expression on primitive surfaces
and halfspaces, the expression can be used as a persistent name for the
cells. In fact, it appears that some commercial systems use this method of
naming in simple cases with limited success.11 Boolean set-theoretic representations place no restrictions on the topology of the resulting boundary
representations, imply a very different parametric family [Shapiro and
Vossler 1995], and do not provide a mechanism for enforcing or even
checking BR-deformation.
A common method of matching cells in pre-edit and post-edit b-reps,
which can be observed in some commercial systems, combines Boolean
naming with the sequential ordering of the entities with respect to some
external direction. For example, a system may attempt to distinguish
between edges e 1 and e 2 in Figure 2(a) by indexing them along the length of
the face (the x-axis in Figure 2). This method is most likely responsible for
producing the solid in Figure 2(d) instead of (c): the identification of e 1 with
e 3 changes the adjacency information so that e92 appears to be the “first”
edge to fit the new name of e 1 .
5. CONCLUSION
5.1 Significance and Limitations
We have argued that the proposed principle of continuity and the implied
notion of BR-deformation should be accepted as the basis for formally
defining the semantics of a parametric family. Notice, however, that
BR-deformation is not an equivalence relation: it is generally not true that
11
Constructing Boolean expressions for cells in a boundary representation are not trivial and
may require a difficult construction known as separation [Shapiro and Vossler 1993].
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boundary representation K can be deformed into L and vice versa. Thus, a
parametric family of boundary representations can be defined in more than
one way. For example, we could define a parametric family to include all
those boundary representations that can be obtained by deforming one
special “master” boundary representation K. Alternatively, we could define
two boundary representations to be in the same parametric family whenever one of them can be deformed into another. The latter definition may be
more appealing because it does specify an equivalence class of solids, but
deciding membership in such a class algorithmically appears to be nontrivial.
The proposed formalism allows precise formulation and partial solution
for a number of problems in parametric modeling. For example, the
problem of “persistent naming” amounts to deciding whether two boundary
representations K and L can be related by a BR-deforming map g that is
consistent with the semantics of the parametric edit, and constructing such
a map.
A key computational utility for enforcing BR-deformation is the ability to
decide if a given naming map g between two b-reps K and L is BRdeforming. We have argued that any such g must satisfy the two easily
computable necessary conditions for every named (mapped) cell s:
—the combinatorial adjacency (star or closure) conditions expressed by (1)
and (2); and
—the orientation condition g p21 (­( s )) 5 ­( g p ( s )).
Our formulations and the implied conditions do not completely solve the
preceding problem, but they are the strongest possible, in the sense that
the formulated problem could be solved only if solid modeling representations are enhanced with additional information that is not being used
today. The main difficulty lies in the need to know what happens to the
boundary representation not only for parameter values t 0 and t 1 , but also
for all the values of t in the interval [t 0 , t 1 ]. For example, it is impossible to
decide that the two boundary representations in Figure 5 do not belong to
the same parametric range without considering the deformation process in
E 3 . This is the very reason why we defined BR-deformation as a family of
maps F t from E 3 into E 3 , even though the naming map g 5 F t 1 takes uKu
into uLu. To guarantee that a given naming map g is BR-deforming, we
must show that g is an orientation-preserving cell map and that it can be
extended to a continuous map on the whole of E 3 . To be more precise, g will
satisfy all the requirements of a BR-deforming map if there exists a
continuous orientation-preserving map h from E 3 to E 3 , such that h
restricted to uKu is a cell map g from uKu to uLu [Agoston 1976].
This sufficient condition for BR-deformation cannot be verified by considering only the structure of the two-dimensional boundary representations.
Possible approaches to constructing BR-deforming maps include constructing the corresponding map h as a cell map on the three-dimensional
decomposition of E 3 as suggested in Shapiro [1997] and Shapiro and
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Boundary Representation Deformation
Fig. 8.
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A parametric edit is defined by a splitting followed by a collapse.
Vossler [1995], or ensuring the properties of h by further restricting how g
modifies individual cells [Andersson et al. 1998; Stewart 1992].
Other criteria for membership in a parametric family are possible. For
example, one could require only that two boundary representations are
related by an orientation-preserving cell map. This would significantly
enlarge the parametric family and would immediately establish computable
sufficient conditions for persistent naming. These advantages would come
at a considerable sacrifice: the loss of the physical principle of continuous
deformation between solids in the same family. Nevertheless, this may be a
reasonable and pragmatic compromise, given the difficulties of establishing
BR-deformation as defined in Section 2.
5.2 Semantics of More General Edits
It may appear that accepting the principle of continuity and BR-deformation as a basis for parametric modeling is too limiting because a number of
practically important parametric edits cannot be described by BR-deformation. These include merging and splitting entities in boundary representations, (dis)appearance of holes, and common edits such as shown in Figure
8: as the slot moves to the left, the edge e 1 collapses into vertex v 1 . But this
edit cannot be described by a cell map unless the edge bounded by vertices
v 1 and v 3 is split into two edges, e 2 and e 3 , as shown in Figure 8(a). This
would allow constructing a proper cell map that takes v 5 into v92 , v 4 into
v91 , and so on. But of course, vertex v 2 would not be present in most
boundary representations that rely on maximal faces [Silva 1981]. In the
following we explain how our approach may be extended to formally define
arbitrary parametric edits.
Although the notion of BR-deformation is not applicable to splitting and
merging of faces and edges (such as needed in the example of Figure 8),
recall that the main purpose of an orientation-preserving cell map was to
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Fig. 9.
S. Raghothama and V. Shapiro
Holes can be eliminated as a two-step procedure: collapse followed by a merge.
enforce the continuity of the map g: uKu 3 uLu in a cell-by-cell fashion; a
natural extension of this principle would allow structural changes in the
cells of K and L. In particular, subdividing or merging cells in a solid’s
boundary representation, for example, as proposed in O’Connor and Rossignac [1990], has no effect on the boundary set itself (the underlying space).
Thus, it seems reasonable that we can define a more general edit that
preserves the spirit of continuous deformation by a sequence of splits,
merges, and BR-deforming maps. It is important to note that such edits
include significant topological changes, including elimination of holes.
Figure 9 illustrates a possible two-step procedure. First the collapsing map
g takes the boundary of the hole c to a vertex c9 5 g(c); the resulting cell
complex has a 2-cell that contains an isolated 0-cell in its interior. Then,
the subsequent merging step eliminates the 0-cell, without changing the
underlying space.
Finally, recall that the expanding map is not BR-deforming (only its
inverse is), because it is one-to-many and is not a continuous cell map. Yet
it appears that useful parametric edits require using such a map. It may be
feasible to define such edits by a sequence of maps such that each map in
the sequence is collapsing, expanding, splitting, or merging. This approach
may provide great flexibility in formally defining the semantics of parametric edits in a piecewise continuous fashion. For example, the reverse
sequence of events shown in Figure 9 would allow the introduction of holes
as a two-step procedure consisting of a split followed by an expanding map.
This may also allow formally defining the semantics of feature attachment
operations without using Boolean operations.
5.3 Deformations of Other Representations
The assumptions of continuity and continuous deformations make sense
from an engineering point of view and can serve as a starting point for
developing universal standards in parametric modeling. The proposed
definition of BR-deformation and the implied necessary conditions can be
employed in numerous other applications, including shape optimization,
tolerancing, and constraint solving. For example, some heuristic methods
proposed in Bouma et al. [1995] for identifying the correct solutions in
constraint solving resemble the necessary conditions implied by our definitions, and the use of homotopy was also proposed in Lamure and Michelucci
[1995].
The principle of continuity is stated in terms of a particular representation scheme for solids. The same principle of continuity could be used to
develop notions of parametric families with respect to other representation
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schemes. For example, it may be feasible to define a parametric family with
respect to CSG representations [Shapiro and Vossler 1995], in which case
the Boolean naming [Rossignac 1996] may indeed correspond to a class of
appropriately defined continuous deformations.
The obvious and useful extension of our work is to general n-dimensional
cellular models of pointsets, such as those represented by Selective Geometric Complexes [O’Connor and Rossignac 1990] and more recently advocated by others [Armstrong et al. 1997; Rappoport 1997]. It should be
intuitively clear that such cellular structures can be constructed and
transformed into each other by combinations of expanding, collapsing,
splitting, and merging maps. The cellular data structures (including boundary representations) are commonly supported by a number of so-called
Euler operators [Braid et al. 1980; Mantyla 1988] that create, modify, and
eliminate cells or collection of cells (loops, shells, rings, wedges, etc.).
Although the Euler operators are assumed to operate continuously on the
underlying space of the cell complex, the continuity conditions are rarely
enforced. The concepts of continuity, orientation-preserving cell map, continuous deformation, and chain map apply to all cellular models without
any modifications. Furthermore, the identified necessary conditions for
deformation, including the star condition (2) and the orientation condition
(3), hold as well.
ACKNOWLEDGMENTS
The authors are grateful to Tom Peters, Malcolm Sabin, Neil Stewart, and
Victor Zandy for reading the earlier drafts of this article and suggesting
numerous improvements.
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Received April 1997; revised April 1998; accepted September 1998
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