Boolean operations in solid modeling: Boundary evaluation and merging algorithms

Requicha, Aristides A. G.; Voelcker, H.

doi:10.1109/proc.1985.13108

Cited by 241 publications

(114 citation statements)

References 21 publications

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“…For example, (w,, y, z) = (O, -6, a) is always a nontrivial solution of (18) This means that the resultant RX, a polynomial in x, has infinitely many roots and thus is identically zero. But notice that in this case (18) could have been homogenized as (19), which is a system of homogeneous equations of degrees 221. In fact, there is a correspondence between rank deficiencies of matrix (8) and degree deficiencies of the homogenized quadric equations.…”

Section: Identically Zero Resultantsmentioning

confidence: 99%

“…For ranks 2, 1, and O, the homogenized quadric equations are of degrees 221, 211, and 111, respectively. They are given by (19), (20), and (21) …”

Section: Identically Zero Resultantsmentioning

confidence: 99%

“…These x roots are extraneous roots; they can be found by solving for those x for which the system of equations derived by setting WXto O in (11), (19), (20), or (21) has nontrivial common solutions. For (11) Equations (26) and (27) are obvious.…”

Section: Rootsmentioning

confidence: 99%

“…The Boundary Evaluation algorithm operating on solids A and B can be summarized at a high level as follows [19]:…”

Section: Introductionmentioning

confidence: 99%

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Using multivariate resultants to find the intersection of three quadric surfaces

1991

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Macaulay's concise but explicit expression for nmltivariate resultants has many potential applications in computer-aided geometric design. Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces. By B6zout's theorem, three quadric surfaces have either at most eight or intlnitely many intersections. Our method finds the intersections, when there are finitely many, by generating a polynomial of degree at most eight whose roots are the intersection coordinates along an appropriate axis. Only addition, subtraction, and multiplication are required to find the polynomial. But when there are pmsibilities of extraneous roots, division and greatest common divisor computations are necessary to identify and remove them.

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Section: Identically Zero Resultantsmentioning

confidence: 99%

“…For ranks 2, 1, and O, the homogenized quadric equations are of degrees 221, 211, and 111, respectively. They are given by (19), (20), and (21) …”

Section: Identically Zero Resultantsmentioning

confidence: 99%

Section: Rootsmentioning

confidence: 99%

“…The Boundary Evaluation algorithm operating on solids A and B can be summarized at a high level as follows [19]:…”

Section: Introductionmentioning

confidence: 99%

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Using multivariate resultants to find the intersection of three quadric surfaces

1991

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“…A large number of research papers have been written about Breps. Most of these papers have dealt with the design and analysis of specific Brep data structures [Woo85,Mäntylä88,Alla91,Guibas85] and with algorithms operating on Breps, such as Boolean operations [Requicha85,Hoffmann89]. The essence of a Brep is best described as explicitly representing open, dimensionally uniform, connectivity components of intersection entities generated by a set of geometric pointset carriers.…”

Section: Introductionmentioning

confidence: 99%

Breps as Displayable-Selectable Models in Interactive Design of Families of Geometric Objects

Rappoport

1997

Geometric Modeling: Theory and Practice

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Abstract. In classical geometric modeling, the primary objects of interest are geometric pointsets, the primary representation scheme for which is the boundary representation (Brep). Modern geometric modeling focuses on parametric families of pointsets, defined using geometric operation graphs (GOGs), features and constraints. During interactive design of families of objects, users interact with an example object from the family. The example object is a pointset, hence is usually modeled using the Brep. In this paper we study the issue of which modeling scheme is most appropriate for the example object. We identify two major operations which such a modeling scheme must support: display and selection. Selection can be further decomposed into picking, invariant naming, and persistent naming. We introduce the term Displayable-Selectable Models (DSmodels) as a generic term for models providing display and selection functionality. We discuss the suitability of Breps to serve as DS-models and whether other, perhaps simpler, representations could also serve as DS-models.

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Solid and Physical Modeling

Rossignac

2007

Wiley Encyclopedia of Electrical and Electronics Engineering

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The sections in this article are Introduction Topological Domain Geometric Domain and Representation Schemes Triangle Meshes Curved BReps Constructive Solid Geometry Parameters, Constraints, and Features Variational Geometry Morphological Transformations and Analysis Human‐Shape Interaction Conclusions

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Boolean operations in solid modeling: Boundary evaluation and merging algorithms

Cited by 241 publications

References 21 publications

Using multivariate resultants to find the intersection of three quadric surfaces

Using multivariate resultants to find the intersection of three quadric surfaces

Breps as Displayable-Selectable Models in Interactive Design of Families of Geometric Objects

Solid and Physical Modeling

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