C. M. Hoffmann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):50--59, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an important task either to evaluate the B rep of a CSG tree, or to compute the union, intersection or di erence between solids in B rep. This question has been tackled many times [1, 2, 3, 4] but problems still arise for a reliable and complete de nition and for the robustness of the operations [5, 6]. To solve the rst kind of problems, we study a new approach based on the re nement of embedded combinatorial 3 maps and on the use of formal methods. Our paper focuses on boolean operations between solids in B rep, even if this work applies to general CSG when the primitives of the CSG trees are ....

....solids in B rep, even if this work applies to general CSG when the primitives of the CSG trees are given in B rep. The robustness problems are mainly rooted in approximation and round o errors due to the oating point arithmetic used in the calculations. To solve them, Ho mann and Hopcroft [5] propose to limit the redundancy of numerical data, considering only plane equations, and add to the geometrical tests a symbolic reasoning system insuring against geometrical decisions that are in contradiction with previous ones. Other purely numerical approaches aim to control and limit the ....

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C.M. Homann, J.E Hopcroft, and M.S. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics & Applications, 9(6):50-59, 1989.

....have been shown to have serious shortcomings, since they may cause not only numerical errors but also fatal membership errors (such as inclusion of a point in an interval to which it does not belong, etc. This diOEculty has received some deserved attention in recent years (see, e.g. [For89, GY86, HHK89, Mil88a, Mil88b, Mil89, GSS89, SI88]) and several approaches have been proposed on how to obviate the shortcoming. The common objective is to produce robust algorithms, namely algorithms whose answer is a (small) perturbation of the cor rect answer (as produced by the innite precision algorithm) As noted by Fortune [For89] there ....

.... by Fortune [For89] there are basically two categories of approaches to this objective: The most common one resorts to approximate (i.e. rounded) computations, and uses properties of the assumed primitives to establish the topological correctness of the results (i.e. robustness) see, e.g. [For89, For92, FM91, HHK89, GSS89]) The other uses ex act (i.e. integer) computations, but, since multiprecision integer arith metic is required (d fold for a dimension d determinant) a straightfor ward implementation of this approach has a large performance penalty (see [FV93] for a detailed analysis) Signicant improvements ....

C. M. Hooemann, J. E. Hopcroft, and M. T. Karasick. Ro bust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):5059, November 1989.

....avoiding degenerate cases can solve most of the problems. However, degenerate cases are often assumed intentionally in geometric modeling, therefore avoiding them may not be appropriate for it possibly fails to capture the original design intention. Symbolic reasoning is used in many approaches[14][16] 19] 20] 21] 30] 31] 37] to maintain the consistency among all the decisions made with regards to geometric relations. The success of reasoning seems to largely depend on the availability of powerful and efficient symbolic reasoning approaches. In the extreme, general geometric theorem proving ....

....and it is updated for every relation computed. These tolerance based approaches update the tolerances according to the decisions made in order to maintain the consistency of the decisions, or to possibly detect an inconsistency among them. Yet another approach described, for instance, in [14] and [39] aims at limiting or elim3 inating redundant data and thus avoids inconsistencies among dependent decisions. Some redundancies can also be directly detected an eliminated in a CSG representation[34] 26] Overview Although geometric modeling algorithms such as those in computational ....

Hoffmann, C. M., Hopcroft, J. E., and Karasick, M. S. Robust set operations on polyhedral solids. IEEE Computer Graphics and Application 9 (November 1989).

....with numerical error is based on using tolerances with floating point arithmetic [15] however it is hard to decide a global tolerance value for all computations. Other approaches for dealing with numerical error include adaptive tolerances [34] and interval arithmetic [14] Symbolic reasoning [12] and redundancy elimination [8] are among the other methods used to increase robustness in solid modeling applications. B rep computation algorithms involve accurate evaluation of the sign of arithmetic expressions, which can present problems for floating point arithmetic when the value of the ....

C. Hoffmann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. ieeeCGA, 9(6):50--59, 1989.

....patterns. A variety of techniques have been designed to make geometric algorithms robust in the presence of high precision numerical computations (e.g. involving square roots) and degenerate geometric configurations (e.g. more than two collinear points or more than three cocircular points) [3, 14, 24, 28, 33, 34, 43, 47, 48, 49, 57, 76, 78, 85, 86, 87]. GeomLib adopts the paradigm of exact computation (see, e.g. Refs. 3, 14, 86] and uses the concept of degree [57] to characterize the arithmetic precision requirement of a geometric algorithm. Namely, a geometric algorithm of degree d requires in its computations a precision that is, in the ....

....the encapsulation of the geometric information within the basic geometric objects allows the implementation of a geometric algorithm to be independent from the arithmetic used. However, the problem of the correctness of the arithmetic computations has to be considered, as indicated, e.g. in Refs. [3, 14, 24, 28, 33, 34, 43, 47, 48, 49, 57, 76, 78, 85, 86, 87]. The assumption of real number arithmetic has proved unrealistic, since digital computers do not exhibit such capability natively, i.e. in hardware. On the other hand, exact rational arithmetic via software may excessively slow down computations. In light of these problems, the equivalent ....

C. M. Ho#mann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):50--59, 1989.

....arithmetic, is minimal. For a generic intersection, there is essentially no overhead. For a contrived hard intersection, the performance cost is less than twice what a floating point modeler might require (though the floating point modeler might well fail because of numerical error [15]) An earlier version of this paper appeared in the Third Symposium on Solid Modeling and Applications [7] Preprint submitted to Elsevier Preprint 1 April 1996 A principal property of the modeler is a bound on the bit length of coordinate data; this implies a fixed, relatively small bound on ....

.... arithmetic for the evaluation of geometric predicates; to reduce performance cost, various researchers have suggested adaptive precision arithmetic [9,16,17] The specific problem of constructing a reliable polygonal or polyhedral modeler has been considered both in floating point arithmetic [4,12,15] and exact arithmetic [1,16,22,27] much of the latter work is discussed in more detail below. a) b) Fig. 1. a) nonmanifold; b) manifold representation. 2 Algorithm design A polyhedral modeler provides boolean set operations on polyhedral solids, as well as affine transformations such as ....

C. Hoffmann, J. Hopcroft, M. Karasick, Robust set operations on polyhedral solids, IEEE Comp. Graph. Appl. 9(6):50--59, 1989.

....and has useful engineering features; for example, scaling is automatic. Of course, floating point does not implement exact real arithmetic. A program may fail because a geometric primitive gives an incorrect answer due to rounding error. Milenkovic[65] and Hoffmann, Hopcroft, and Karasick[56] give clear examples of such failure. Standard error analysis techniques can be applied to geometric primitives[22] Suppose the input to an algorithm is a set of points. The algorithm can interpret the coordinates of the points as being exact, ignoring any error in the process that generated the ....

....be used to answer nearest neighbor queries. If the point location algorithm reports that a query point q is in the region of a Voronoi site s, then s may not be the true closest site, but any other site is closer only by a relative amount proportional to O(nffl) Hoffmann, Hopcroft, and Karasick[56, 58] consider set operations on polyhedral solids using floating point arithmetic. They outline a procedure for intersecting the bounding shells of polyhedral solids in order to minimize the effect of numerical uncertainty. They give an elaborate hierarchy of tests for deciding vertex vertex, ....

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C. Hoffmann, J. Hopcroft, M. Karasick, Robust set operations on polyhedral solids, IEEE Comp. Graph. Appl. 9(6):50--59, 1989.

....a general framework. This framework was first introduced by Requicha and Voelcker [RV85] to perform Boolean operations on polyhedra. However, this can be extended easily to accommodate curve surface domains as well. Given two polyhedra, A and B, the conceptual structure of the algorithm [Hof89, HHK89] is as follows: 1. Determine which pairs of faces f 2 A and g 2 B intersect. If there are none, test for containment only and skip all other steps. 2. For each face f 2 A that intersects a face of B, construct the cross section of B with 6 the plane containing f . Then determine the surface ....

....accurate B rep computation in polyhedral modelers. One of the most common approaches is based on using tolerances with floating point arithmetic [Jac95] However, it is hard to decide a global tolerance value for all computations. To circumvent these problems, combinations of symbolic reasoning [HHK89] and adaptive tolerances [Seg90] have been proposed. Other algorithms include those based on redundancy elimination [FBZ93] Many algorithms based on exact arithmetic have been proposed for reliable numeric computation for polyhedra [SI89, For95, BMP94, Hof89] Sculptured Solids: The idea of ....

C. Hoffmann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications, 9(6):50--59, 1989.

....documented in the literature. Several proposals have been made to remedy this unsatisfactory situation. They can be split into two broad categories according to whether they perform exact computations (see, e.g. BKM 95, FV93, Yap97, She96] or approximate computations (see, e.g. Mil88, HHK89, Mil89] This paper ne tunes the exact computation paradigm. The numerical computations of a geometric algorithm are basically of two types: tests (predicates) and constructions, with clearly distinct roles. Tests are associated with branching decisions in the algorithm that determine the AEow ....

C. M. Hooemann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):5059, November 1989.

....has proved unrealistic and needs to be replaced with a realistic finite precision model where geometric computations can be carried out either exactly or with a guaranteed error bound. This has motivated a great deal of research on the subject of robust computational geometry (see, e.g. [4, 11, 10, 18, 26, 27, 30, 35, 33, 38, 47, 53, 56, 20, 29, 31]) Also, efficiency must be evaluated in a finer framework than the conventional big Oh analysis. In particular, constant factors dependent on the precision requirement of the numerical computations should be taken into account. For an early survey of the different approaches to robust ....

C. M. Hoffmann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):50--59, Nov. 1989.

....likely, but we focus our attention on gaps for ease of exposition. Concept Reality Figure 1: Gaps along Boundary Curve Interoperability for Distributed Design Systems 3 3 Related Work The problem of accuracy in geometric modeling for CAD has attracted considerable attention in the last decade [5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 25, 27, 30], with attention both to errors inherent to floating point arithmetic and to numerical approximations in algorithms. However, the use of associated tolerances within modeling implementations has received scant attention in the literature (Please see, for example [26] compared to their ....

Hoffmann, C. M., Hopcroft, J. E. and Karasick, M. S., Robust set operations on polyhedral solids, IEEE CG & A (9), No. 6, November 1989, 50-59.

....to detect and analyze gaps and even, in some elementary cases, to propose fixes. Prior to the work reported here, no adequate diagnostic tool has existed to assess gaps within STEP. 3 Related Work This CAD modeling robustness problem has attracted considerable attention in the last decade [7, 11, 13, 14, 15, 17, 24, 29], where some of these authors [14, 17, 24] have discussed that a numerical method had solved exactly a problem close to the given problem, as is considered in backward error analysis 8 . The necessary characterization of ill conditioned problems and the development of responsive methods have not ....

....cases, to propose fixes. Prior to the work reported here, no adequate diagnostic tool has existed to assess gaps within STEP. 3 Related Work This CAD modeling robustness problem has attracted considerable attention in the last decade [7, 11, 13, 14, 15, 17, 24, 29] where some of these authors [14, 17, 24] have discussed that a numerical method had solved exactly a problem close to the given problem, as is considered in backward error analysis 8 . The necessary characterization of ill conditioned problems and the development of responsive methods have not yet been completely developed for CAD ....

Hoffmann, C. M., Hopcroft, J. E. and Karasick, M. S., Robust set operations on polyhedral solids, IEEE CG & A (9), No. 6, November 1989, 50-59.

....have been shown to have serious shortcomings, since they may cause not only numerical errors but also fatal membership errors (such as inclusion of a point in an interval to which it does not belong, etc. This difficulty has received some deserved attention in recent years (see, e.g. [For89, GY86, HHK89, Mil88a, Mil88b, Mil89, GSS89, SI88]) and several approaches have been proposed on how to obviate the shortcoming. The common objective is to produce robust algorithms, namely algorithms whose answer is a (small) perturbation of the correct answer (as produced by the infinite precision algorithm) As noted by Fortune [For89] there ....

.... by Fortune [For89] there are basically two categories of approaches to this objective: The most common one resorts to approximate (i.e. rounded) computations, and uses properties of the assumed primitives to establish the topological correctness of the results (i.e. robustness) see, e.g. [For89, For92, FM91, HHK89, GSS89]) The other uses exact (i.e. integer) computations, but, since multiprecision integer arithmetic is required (d fold for a dimension d Thetad determinant) a straightforward implementation of this approach has a large performance penalty (see [FV93] for a detailled analysis) Significant ....

C. M. Hoffmann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):50--59, November 1989.

....to fail on the input. This problem is often referred to as the robustness problem [24] One solution to the robustness problem is to explicitly handle numerical inaccuracies, so as to design an algorithm that does not fail even if the numerical part of the computation is done approximately [25, 37], or to analyze the error due to the f.p. imprecision [19] Such designs are extremely involved and have only been done for a few algorithms. The general solution, it has been widely argued, is to compute certain predicates exactly [11, 15, 17, 20, 41] see also section 6.2) This is also the ....

C. M. Hooemann, J. E. Hopcroft, and M. T. Karasick, Robust set operations on polyhedral solids, IEEE Comput. Graph. Appl. 9:6 (1989) 5059.

....based on using tolerances with floating point arithmetic [Jac95] If two geometric elements are within the given tolerance, they are considered incident. However, it is hard to decide a global tolerance value for all computations. To circumvent these problems, combinations of symbolic reasoning [HHK89] and adaptive tolerances [Seg90] have been proposed. Other algorithms include those based on redundancy elimination [RV89, FBZ93] B rep computation algorithms involve accurate evaluation of the sign of arithmetic expressions. Algorithms based on floating point arithmetic are at times ambiguous, ....

C. Hoffmann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications, 9(6):50--59, 1989.

....proposed for robust and accurate B rep computation. One of the most common approaches is based on using tolerances with floating point arithmetic [Jac95] however it is hard to decide a global tolerance value for all computations. To circumvent these problems, combinations of symbolic reasoning [HHK89] and adaptive tolerances [Seg90] have been proposed. Other algorithms include those based on redundancy elimination [FBZ93] B rep computation algorithms involve accurate evaluation of the sign of arithmetic expressions, which can present problems for floating point arithmetic when the value of ....

C. Hoffmann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications, 9(6):50-- 59, 1989.

.... has also created line arrangement and plane arrangement algorithms that satisfy this definition [14] For a discussion of the more general problem of improving the reliability geometric computations using rounded arithmetic, see [8, 9] Problems addressed include polygon [19, 13] and polyhedron [10, 11, 20] modeling and the calculation of convex hulls [17] and other problems in plane geometry [15] Karasick [11] has the most practical result to date, an ultra reliable polyhedral modeling system that is provably safe: it will not generate a topologically inconsistent result. Greene and Yao [7] have ....

Christoph M. Hoffmann, John E. Hopcroft, and Michael S. Karasick. Robust Set Operations on Polyhedral Solids. Technical Report 87-875, Department of Computer Science, Cornell University, Ithaca, New York 14853-7501, October 1987.

.... This problem is often referred to as the robustness problem in computational geometry [17] One solution to the robustness problem is to explicitly handle numerical inaccuracies, so as to design an algorithm that does not fail even if the numerical part of the computation is done approximately [19, 29], or to analyze the error due to the f.p. imprecision [12] Such designs are extremely involved and have only been done for a few algorithms. The general solution, it has been widely argued, is to compute the predicates exactly [9, 6, 13, 33, 11] This can be achieved in many ways: computing the ....

C. M. Hooemann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):5059, November 1989.

.... Euclidean space, and set theoretic operations are replaced by their regularized versions [22] Several algorithms exist for solving Boolean operations on polyhedral solids, but not all satisfactorily address the crucial problem of numerical errors that are inherent to floating point computations [7, 10, 11, 19, 20, 21]. In this paper, we present solutions that we have experimented in the implementation of a solid modeler. Our approach is based on a general algorithm and data structure that naturally accommodate non manifold geometric cases. Numerical errors are avoided by the use of a new kind of exact ....

....Segal and Sequin [18, 19, 20] impose a minimum separation between each pair of primitives (faces, edges, or vertices) any two primitives that are within less than a chosen minimal distance must be either merged or pulled apart to maintain the minimum separation. Hoffmann, Hopcroft and Karasick [7] also impose a minimum separation, but they resort to an additional symbolic reasoning to ensure a decision never contradicts previous ones. Milenkovic [15] also places a higher priority on topology. He describes a verifiable implementation of a line arrangement algorithm, that maintains a ....

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Hoffmann, C M, Hopcroft, J E and Karasick, M S. Robust Set Operations on Polyhedral Solids. IEEE CG&A, 9( 6), Nov. 1989, pp. 50-59.

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C. M. Hoffmann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9(6):50--59, 1989.

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Hoffmann C, Hopcroft J, Karasick M. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications 1989;9(6):50--9.

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C. Hoffmann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications, 9(6):50--59, 1989.

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#3#:31#41, 1989. #33# C. M. Ho#mann, J. E. Hopcroft, and M. T. Karasick. Robust set operations on polyhedral solids. IEEE Comput. Graph. Appl., 9#6#:50#59, 1989. #34# J. E. Hopcroft and P. J. Kahn. A paradigm for robust geometric algorithms. Algorithmica, 7#4#:339#380,

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C. Ho#mann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications, 9#6#:50#59, 1989.

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C. Hoffmann, J. Hopcroft, and M. Karasick. Robust set operations on polyhedral solids. IEEE Computer Graphics and Applications, 9(6):50--59, 1989.

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