## Active zones in CSG for accelerating boundary evaluation, redundancy elimination, interference detection, and shading algorithms (1989)

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Venue: | ACM Transactions on Graphics |

Citations: | 27 - 8 self |

### BibTeX

@ARTICLE{Rossignac89activezones,

author = {Jaroslaw R. Rossignac and Herbert B. Voelcker},

title = {Active zones in CSG for accelerating boundary evaluation, redundancy elimination, interference detection, and shading algorithms},

journal = {ACM Transactions on Graphics},

year = {1989},

volume = {8},

pages = {51--87}

}

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### Abstract

Solids defined by Boolean combinations of solid primitives may be represented in constructive solid geometry (CSG) as binary trees. Most CSG-based algorithms (e.g., for boundary evaluation, graphic shading, interference detection) do various forms of set-membership classification by traversing the tree associated with the solid. These algorithms usually generate intermediate results that do not contribute to the final result, and hence may be regarded as redundant and a source of inefficiency. To reduce such inefficiencies, we associate with each primitive A in a tree S an active zone 2 that represents the region of space where changes to A affect the solid represented by S, and we use a representation of 2 instead of S for set-membership classification. In the paper we develop a mathematical theory of active zones, prove that they correspond to the intersection of certain nodes of the original trees, and show how they lead to efficient new algorithms for boundary evaluation, for detecting and eliminating redundant nodes in CSG trees, for interference (null-set) detection, and for graphic shading.

### Citations

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Citation Context ...aint-based specifications in the CSG representation, and Peterson [16] and Vossler [37] described techniques for converting 2-D contours into CSG representations of 2-D regions and 3-D swept volumes. =-=(3)-=- Most CSG-based systems were developed for representing and manipulating homogeneously 2-D regions or 3-D solids. Certain CAD/CAM applications, however, deal with multi-dimensional collections of open... |

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Citation Context ... A solid (a) can be represented as A + B (b), A + C (c), D - F(d), and so on. CSG. Other applications require that some form of partial or complete boundary eualuation be done. Requic:ha and Voelcker =-=[20]-=- described the basic techniques for boundary evaluation, which is inherently computationally intensive, and Woodwark and Quinlan [40] (and many others) proposed improvements to speed up boundary evalu... |

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Citation Context ...from S to A as - nodes. Consequently, classifying candidate elements against I, U, and 2 can be done by using a slightly modified version of set-membership classification algorithms developed for CSG =-=[32]-=-. We derive below some useful properties of I, II, and 2, and prove that Z (defined as I - U) is the region where changes to A affect S. An intuitive understanding of this property can be gained from ... |

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Citation Context ...sometimes impossible, to specify certain well-defined operations (such as offsetting and blending) in these terms. Rossignac and Requicha added constant radius blending [24] and offsetting operations =-=[25]-=- to the CSG scheme. Shirma et al. [30] studied tools for modeling Boolean combinations of sweeps, and Van Wijk [36] developed tools for displaying them. (2) The design in CSG of complex 3-D solids req... |

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Citation Context ...developed for representing and manipulating homogeneously 2-D regions or 3-D solids. Certain CAD/CAM applications, however, deal with multi-dimensional collections of open geometric elements. Cameron =-=[5]-=- used 4-D CSG representations to compute intersections of moving objects, and Rossignac and O’Connor [23] proposed algorithms for performing Boolean operations on nonhomogeneous, not necessarily close... |

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Citation Context ... S, and B be w-redundant in S(A t w). Substituting w and substituting w for B in (10) yields S = (B + K) . M, S = M. Combining (9) and (11) yields M C A + J, and therefore, (10) (11) M = (A + J) . M, =-=(12)-=- implying that A is w-redundant in S(B t w), which is equal to (A + J) . M. Substituting (12) in (11) yields S = (A + J) + M, (13) implying that B is w-redundant in S. We have shown that “A w-redundan... |

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Citation Context ...ions on Graphics, Vol. 8, No. 1, January 1989.s74 - J. FL Rossignac and H. 8. Voelcker implying that A is 0-redundant in S(B t 0), which is equal to A . J + M. Substituting (7) in (6) yields S=A.J+M, =-=(8)-=- implying that B is O-redundant in S. We have shown that “A 0-redundant in S, and B 0-redundant in S(A t 0)” implies “B 0-redundant in S, and A 0redundant in S(B t 0).” The converse implication is tru... |

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Citation Context ...us blending [24] and offsetting operations [25] to the CSG scheme. Shirma et al. [30] studied tools for modeling Boolean combinations of sweeps, and Van Wijk [36] developed tools for displaying them. =-=(2)-=- The design in CSG of complex 3-D solids requires that the users produce sequences of rotations and translations that will correctly position the solid primitives. This task is particularly hard when ... |

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Citation Context ...ations to compute intersections of moving objects, and Rossignac and O’Connor [23] proposed algorithms for performing Boolean operations on nonhomogeneous, not necessarily closed, n-dimensional sets. =-=(4)-=- A CSG representation of a given solid is not unique (e.g., see Figure 2), and thus, it is expensive to determine whether two CSG trees represent the same solid and whether any particular CSG tree rep... |

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Citation Context ...slations that will correctly position the solid primitives. This task is particularly hard when it is necessary to achieve complex 3-D arrangements of primitives bounded by curved surfaces. Rossignac =-=[22]-=- and Anderson [l] incorporated constraint-based specifications in the CSG representation, and Peterson [16] and Vossler [37] described techniques for converting 2-D contours into CSG representations o... |

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Citation Context ...sents rigorously the same Boolean function and thus the same solid as the original tree. The three regularized operators, union, intersection, and complement, form a Boolean algebra over regular sets =-=[18]-=-. Therefore, De Morgan’s laws [28] can be invoked to manipulate CSG expressions. Any CSG tree may be converted into its positive form by a preorder traversal of the tree applying, when appropriate, th... |

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Citation Context ...Rossignac and Requicha added constant radius blending [24] and offsetting operations [25] to the CSG scheme. Shirma et al. [30] studied tools for modeling Boolean combinations of sweeps, and Van Wijk =-=[36]-=- developed tools for displaying them. (2) The design in CSG of complex 3-D solids requires that the users produce sequences of rotations and translations that will correctly position the solid primiti... |

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Citation Context ...al or complete boundary eualuation be done. Requic:ha and Voelcker [20] described the basic techniques for boundary evaluation, which is inherently computationally intensive, and Woodwark and Quinlan =-=[40]-=- (and many others) proposed improvements to speed up boundary evaluation. The generation of shaded pictures from CSG has received a great deal of attention; many references may be found in [lo] and [2... |

16 | Efficient intersection tests for objects defined constructively
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Citation Context ...lid and whether any particular CSG tree represents the empty set. Tilove [33, 341 dealt with efficient .methods for comparing sets represented by two different (possibly empty) CSG trees, and Cameron =-=[6]-=- proposed improvements to some of Tilove’s results. Woodwark [39] described a different method, which simplifies CSG representations by reasoning on geometric approximations of combinations of half-sp... |

16 |
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Citation Context ... Null-Object Detection. Null (empty) solids provide a blatant example of redundancy, because they may have arbitrarily elaborate CSG representations in which all primitives and subtrees are redundant =-=[34]-=-. Null-object ACM Transactions on Graphics, Vol. 8, No. 1, January 1989.s56 l J. R. Rossignac and H. 6. Voelcker detection (abbreviated NOD) is an important problem in solid modeling. Good examples ar... |

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Citation Context ... S, and B be 0-redundant in S(A t 0). Substituting 0 for A in (4) yields and substituting 0 for B in (5) yields S=B.K+M, (5) S = M. (6) Combining (4) and (6) yields A . J C M, and therefore, M=A.J+M, =-=(7)-=- ACM Transactions on Graphics, Vol. 8, No. 1, January 1989.s74 - J. FL Rossignac and H. 8. Voelcker implying that A is 0-redundant in S(B t 0), which is equal to A . J + M. Substituting (7) in (6) yie... |

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Citation Context ... in terms of A and B. Let A be w-redundant for A in (9) yields in S, and B be w-redundant in S(A t w). Substituting w and substituting w for B in (10) yields S = (B + K) . M, S = M. Combining (9) and =-=(11)-=- yields M C A + J, and therefore, (10) (11) M = (A + J) . M, (12) implying that A is w-redundant in S(B t w), which is equal to (A + J) . M. Substituting (12) in (11) yields S = (A + J) + M, (13) impl... |

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Citation Context ...0] (and many others) proposed improvements to speed up boundary evaluation. The generation of shaded pictures from CSG has received a great deal of attention; many references may be found in [lo] and =-=[26]-=-. 1.3 Specific Problems Addressed in this Paper This paper focuses mainly on the last two points and seeks to improve the performance of boundary evaluation and several other CSG-based algorithms by r... |

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Citation Context ...mitive redundancy and leads to improvements of the performance of redundancy-detection algorithms. Our algorithms may be used in conjunction with the spatial decomposition techniques proposed in [4], =-=[33]-=-, and [39]. 1.3.3 Improve Null-Object Detection. Null (empty) solids provide a blatant example of redundancy, because they may have arbitrarily elaborate CSG representations in which all primitives an... |

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Citation Context ...rther processing, may be parsed and stored as a binary tree, called the CSG (constructive solid geometry) representation of the resulting solid (Figure 1). Some modelers, for example TIPS [15], BUILD =-=[9]-=-, and GDP [38], represent solids by N-way trees, where each node may have more than two children. Results described in this paper also apply to such representations, because the N children of a node m... |

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Citation Context ...sary to achieve complex 3-D arrangements of primitives bounded by curved surfaces. Rossignac [22] and Anderson [l] incorporated constraint-based specifications in the CSG representation, and Peterson =-=[16]-=- and Vossler [37] described techniques for converting 2-D contours into CSG representations of 2-D regions and 3-D swept volumes. (3) Most CSG-based systems were developed for representing and manipul... |

7 |
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Citation Context ...omplex 3-D arrangements of primitives bounded by curved surfaces. Rossignac [22] and Anderson [l] incorporated constraint-based specifications in the CSG representation, and Peterson [16] and Vossler =-=[37]-=- described techniques for converting 2-D contours into CSG representations of 2-D regions and 3-D swept volumes. (3) Most CSG-based systems were developed for representing and manipulating homogeneous... |

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Citation Context ...n algorithms [5] essential in robotics, and in the automatic verification of machining processes, where both invasive machining andgoal attainment can be tested through the detection of null objects4 =-=[31]-=-. Same-object detection ,tests, such as “Is P equal R?,” can clearly be expressed as a NOD test on the symmetric difference (P - R) + (R - P), or equivalently on P + R - P . R. A standard approach to ... |

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Modeling for Production Design
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Citation Context ...ing, may be parsed and stored as a binary tree, called the CSG (constructive solid geometry) representation of the resulting solid (Figure 1). Some modelers, for example TIPS [15], BUILD [9], and GDP =-=[38]-=-, represent solids by N-way trees, where each node may have more than two children. Results described in this paper also apply to such representations, because the N children of a node may be processe... |

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Citation Context .... Tilove [33, 341 dealt with efficient .methods for comparing sets represented by two different (possibly empty) CSG trees, and Cameron [6] proposed improvements to some of Tilove’s results. Woodwark =-=[39]-=- described a different method, which simplifies CSG representations by reasoning on geometric approximations of combinations of half-spaces. (5) A CSG representation, although informationally complete... |

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Citation Context ... others are transitory and dissolve gradually as our understanding of CSG and its associated algorithms grows. We list below some perceived problem areas together with examples of pertinent research: =-=(1)-=- Practical implementations of CSG restrict the represented objects to be semialgebraic sets specified as regularized Boolean combinations of algebraic ’ A regular set is equal to the topological closu... |

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Citation Context ...rdering and factoring the conjunctive normal form of S in terms of A and B. Let A be w-redundant for A in (9) yields in S, and B be w-redundant in S(A t w). Substituting w and substituting w for B in =-=(10)-=- yields S = (B + K) . M, S = M. Combining (9) and (11) yields M C A + J, and therefore, (10) (11) M = (A + J) . M, (12) implying that A is w-redundant in S(B t w), which is equal to (A + J) . M. Subst... |

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Citation Context ... and (11) yields M C A + J, and therefore, (10) (11) M = (A + J) . M, (12) implying that A is w-redundant in S(B t w), which is equal to (A + J) . M. Substituting (12) in (11) yields S = (A + J) + M, =-=(13)-=- implying that B is w-redundant in S. We have shown that “A w-redundant in S, and B w-redundant in S(A t w)” implies “B w-redundant in S, and A Wredundant in S(B t w).” The converse implication is tru... |

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Citation Context ... of the spatial position of the primitives. The concepts reported in this paper have been successfully implemented by the authors in experimental systems for redundancy detection, boundary evaluation =-=[21]-=-, and shaded picture generation [26]. They also provided a theoretical foundation for some improvements of Tilove’s NOD algorithm. APPENDIX. Classification Algorithms The positive form can be obtained... |

1 |
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Citation Context ...lications, however, deal with multi-dimensional collections of open geometric elements. Cameron [5] used 4-D CSG representations to compute intersections of moving objects, and Rossignac and O’Connor =-=[23]-=- proposed algorithms for performing Boolean operations on nonhomogeneous, not necessarily closed, n-dimensional sets. (4) A CSG representation of a given solid is not unique (e.g., see Figure 2), and ... |

1 |
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Citation Context ...function and thus the same solid as the original tree. The three regularized operators, union, intersection, and complement, form a Boolean algebra over regular sets [18]. Therefore, De Morgan’s laws =-=[28]-=- can be invoked to manipulate CSG expressions. Any CSG tree may be converted into its positive form by a preorder traversal of the tree applying, when appropriate, the following transformations to eac... |

1 |
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Citation Context ...in well-defined operations (such as offsetting and blending) in these terms. Rossignac and Requicha added constant radius blending [24] and offsetting operations [25] to the CSG scheme. Shirma et al. =-=[30]-=- studied tools for modeling Boolean combinations of sweeps, and Van Wijk [36] developed tools for displaying them. (2) The design in CSG of complex 3-D solids requires that the users produce sequences... |

1 | Efficient editing of solid models by exploiting structural and spatial locality - G, HOPKINS - 1984 |