## Boundary Representation Deformation in Parametric Solid Modeling (1998)

Venue: | ACM Transactions on Graphics |

Citations: | 19 - 4 self |

### BibTeX

@ARTICLE{Raghothama98boundaryrepresentation,

author = {Srinivas Raghothama and Vadim Shapiro},

title = {Boundary Representation Deformation in Parametric Solid Modeling},

journal = {ACM Transactions on Graphics},

year = {1998},

volume = {17},

pages = {259--286}

}

### Years of Citing Articles

### OpenURL

### Abstract

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...

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Citation Context ...Algebraic topology is also the proper setting for formulating BR-deformation. Below we use a number of standard concepts (complexes, chains, continuous maps, etc.) that can be found in many textbooks =-=[1, 5, 17, 22]-=-. For brevity, we only state few critical definitions and results that are necessary for formulating and understanding the concept of BR-deformation. 2 Blends provide a convenient way to track the nam... |

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Citation Context ...Algebraic topology is also the proper setting for formulating BR-deformation. Below we use a number of standard concepts (complexes, chains, continuous maps, etc.) that can be found in many textbooks =-=[1, 5, 17, 22]-=-. For brevity, we only state few critical definitions and results that are necessary for formulating and understanding the concept of BR-deformation. 2 Blends provide a convenient way to track the nam... |

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Citation Context ...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 =-=[6]-=- 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 [19]. The principle of continuit... |

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Citation Context ... 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. Recently, several schemes for persistent naming have been proposed =-=[10, 18, 25, 31]-=- that appear to work well in a variety of situations. But the proposed techniques are not always consistent between themselves and none come with guarantees or with clearly stated limitations; they ma... |

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Citation Context ... 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, 11, 18, 25, 31]-=- 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 paper, but it is instructive to check if the propo... |

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Citation Context ...Algebraic topology is also the proper setting for formulating BR-deformation. Below we use a number of standard concepts (complexes, chains, continuous maps, etc.) that can be found in many textbooks =-=[1, 5, 17, 22]-=-. For brevity, we only state few critical definitions and results that are necessary for formulating and understanding the concept of BR-deformation. 2 Blends provide a convenient way to track the nam... |

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Citation Context ... 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, 11, 18, 25, 31]-=- 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 paper, but it is instructive to check if the propo... |

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Citation Context ...ng the corresponding map h as a cell map on the three-dimensional decomposition of E 3 as suggested in [32, 34], or ensuring the properties of h by further restricting how g modifies individual cells =-=[4, 36]-=-. 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... |

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Citation Context ...l tolerancing and robustness [29, 36]. Notably, Stewart proposed a definition of variational class in [36], which was later used in study of polyhedral perturbations that preserve topological form in =-=[3]-=-. The assumptions of continuity and continuous deformation are also mentioned informally in the naming techniques and discussion of [11]. Our definitions are consistent with the earlier definitions, b... |

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Citation Context ...y 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 =-=[7, 20]-=- that create, modify, and eliminate cells or collection of cells (loops, shells, rings, wedges, etc.). Though the Euler operators are assumed to operate continuously on the underlying space of the cel... |

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Citation Context ... 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 =-=[11, 13, 15, 34]-=-, and we will 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 defi... |

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Citation Context ... 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 =-=[29, 36]-=-. Notably, Stewart proposed a definition of variational class in [36], which was later used in study of polyhedral perturbations that preserve topological form in [3]. The assumptions of continuity an... |

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Citation Context ...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 =-=[13, 14, 28]-=-. 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... |

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Citation Context ... 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 =-=[11, 13, 15, 34]-=-, and we will 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 defi... |

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Citation Context ...on; 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, e.g. as proposed in =-=[23]-=-, 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... |

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Citation Context ...epresentations. Applying similar principles of continuity to other representations would lead to different definitions of parametric families, e.g. those based on small changes in CSG representations =-=[30, 31, 34]-=-. We do not consider such families in this paper because most extant commercial systems are based on b-reps. 1.3 Examples The following examples illustrate some of the erratic and inconsistent behavio... |

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Citation Context ... 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 =-=[29, 36]-=-. Notably, Stewart proposed a definition of variational class in [36], which was later used in study of polyhedral perturbations that preserve topological form in [3]. The assumptions of continuity an... |

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Citation Context ...bvious and useful extension of our work is to general n-dimensional cellular models of pointsets, such as those represented by Selective Geometric Complexes [23] and more recently advocated by others =-=[8, 26]-=-. 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 da... |

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Citation Context ...allow constructing a proper cell map that takes v 5 into v 0 2 , v 4 into v 0 1 , and so on. But of course, vertex v 2 would not be present in most boundary representations that rely on maximal faces =-=[35]-=-. Below we explain how our approach may be extended to formally define arbitrary parametric edits. (a) Original solid's b-rep K (b) Modified solid's b-rep L v1' v2' v3' e3' e2' e1 e2 v2 v1 e4 e3 v3 v4... |

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Citation Context ... 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. Recently, several schemes for persistent naming have been proposed =-=[10, 18, 25, 31]-=- that appear to work well in a variety of situations. But the proposed techniques are not always consistent between themselves and none come with guarantees or with clearly stated limitations; they ma... |

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Citation Context ...bvious and useful extension of our work is to general n-dimensional cellular models of pointsets, such as those represented by Selective Geometric Complexes [23] and more recently advocated by others =-=[8, 26]-=-. 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 da... |

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Citation Context ...the Cauchy's definition of continuity in terms of open neighborhoods [2]. The combinatorial equivalent of a cell's open neighborhood is the star of the cell. The following definition is modified from =-=[9, 22]-=- and is also consistent with definitions in [23]. Definition 5 (Star of a cell) The star of a cell oe in a complex K, denoted St(oe), is the union of oe and all cells in K that contain oe in their bou... |

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Citation Context ... 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. Recently, several schemes for persistent naming have been proposed =-=[10, 18, 25, 31]-=- that appear to work well in a variety of situations. But the proposed techniques are not always consistent between themselves and none come with guarantees or with clearly stated limitations; they ma... |

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Citation Context ...erfaces 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 =-=[16]-=- 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, an... |

1 |
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Citation Context ...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 : jKj ! jLj on the whole boundary of the solid =-=[1, 22, 24]-=-. By definition, continuity of g requires that g(closure(oe)) ae closure(g(oe)) for every cell oe [21]. But since K and L are finite, and jKj and jLj are closed and bounded, the stronger condition g(c... |