Yap, C. K., Robust geometric computation, in CRC Handbook on Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke (eds.), CRC Press, 1997, 653--668.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for obtaining good resolution bounds, which we anticipate will lead to a better understanding of the method and will open the way to applying the method in other settings. Related work Robustness and precision issues have been intensively studied in Computational Geometry in recent years [24] [27]. A prevailing approach to overcoming robustness issues in Computational Geometry is to use exact computation [20] 28] Such a strategy gives accurate results, and sometimes even allows the input to be degenerate. When applied naively, exact computation can considerably slow down the ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of discrete and computational geometry, pages 653--668. CRC Press, Inc., 1997.

....and hence cannot guarantee the computation of the correct sign. As a consequence, a computation may follow an incorrect path. This may lead to catastrophic errors. In the realm of geometric computations, the situation is particularly severe and known as the precision caused robustness problem [15, 18, 27, 36, 39]. A popular approach to overcoming the robustness problem is the exact computation paradigm [1, 2, 4, 5, 9, 10, 7, 13, 25, 33, 40, 41] The paradigm calls for the exact evaluations of all conditions and hence the exact computation of signs. In this paper, we consider the sign computation for the ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, pages 653--668, CRC Press, 1997.

....under the Real RAM model does not necessarily translate into a robust and or eOEcient program, and catastrophic behaviors are commonly observed. A rst approach to remedy this problem is to use exact arithmetic. In the context of geometric algorithms, much progress has been done in the recent past [19, 20, 16, 14]. Another approach, to be followed here, has emerged recently. Decisions in geometric algorithms depend on geometric predicates which are usually algebraic expressions. For example, for a triple of points given by their Cartesian coordinates, deciding what is the orientation of the triangle ....

C. K. Yap. Robust geometric computation. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653668. CRC Press LLC, Boca Raton, FL, 1997. 11

.... real RAM model of computation and under the assumption of general position, namely that the input is degeneracyfree. These assumptions raise great diculties in implementing robust geometric algorithms. A variety of techniques have been proposed in recent years to overcome these diculties [16] [17]. One approach to robust computing produces a nite precision approximation of the geometric objects in question; for a survey of nite precision approximation algorithms, see, e.g. 15] Snap This work has been supported in part by the IST Programme of the EU as a Shared cost RTD (FET Open) ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, Boca Raton, FL, 1997. 16 Input SR output ISR output Input zoom in SR output zoom in ISR output zoom in

....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. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press, Boca Raton, FL, 1997.

....Two issues that are often ignored in the theoretical approach turn out to be critical in practice: Degeneracies and numerical precision. These issues are collectively referred to as robustness and they have been the topic of extensive research. Surveys on the topic can be found in [24] 27] [36], also several brief state of the art summaries on the topic are collected in [25] In theory degeneracies are often handled by assuming general position, namely assuming that degeneracies do not occur. The general position assumption had contributed significantly to the advancement of geometric ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press LLC, 1997.

....the Real RAM model does not necessarily translate into a robust and or efficient program, and catastrophic behaviours are commonly observed. A first approach to remedy this problem is to use exact arithmetic. In the context of geometric algorithms, much progress has been done in the recent past [16, 17, 14, 12]. Another approach, to be followed here, has emerged recently. Decisions in geometric algorithms depend on geometric predicates which are usually algebraic expressions. For example, for a triple of points given by their cartesian coordinates, deciding what is the orientation of the triangle ....

C. K. Yap. Robust geometric computation. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press LLC, Boca Raton, FL, 1997.

....theory to practice in geometric algorithms. Ignoring these issues can result in unreliable or incorrect programs. This has led to an intensive study in recent years. We brie y mention the main approaches to handling robustness issues next and refer the reader to recent surveys on the topic [69] [79] for further information. Let us illustrate the problem of arithmetic precision in geometric computing. Consider the two polygons on the left hand side of Figure 1: does the small polygon t inside the cavity in the larger polygon We nd the answer by computing the Minkowski sum of the larger ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, Boca Raton, FL, 1997.

....assumption that a solution can be found within the xed precision computation (which is the main computational paradigm in current scienti c computation) For a variety of reasons, these solutions have not been satisfactory. The basis of our approach is the Exact Geometric Computation (EGC) See [10] for a survey. EGC can, in principle, eliminate qualitative errors in the class of algebraic problems. The main challenge is to make EGC solutions e cient, or, e cient enough , so that a user would prefer robust EGC solutions over a fast but nonrobust one. Recent papers Abstract of Invited ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, 1997. 52 chapters, approx. xiv+987 pp.

....may crash or compute garbage. There are two ways to resolve this dilemma: one may either design new algorithms that work correctly even with imprecise arithmetic or implement the real RAM. The number of problems that have been successfully attacked by the rst approach is still small (see [13, 15, 28, 33] for surveys) and the techniques used in the solutions do not obviously generalize to other geometric problems. The main disadvantage of the approach is the necessary redesign; the approach requires to redo computational geometry. The alternative is to base geometric computation on the Real RAM ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, CRC Handbook in Computational Geometry, pages 653-668. CRC Press, 1997.

....[12] on the exact evaluation of integer determinants, Bronnimann et al. 11] on exact geometric predicates with single precision arithmetic, and Shewchuk s design and implementation [43] of four predicates based on adaptive floating point arithmetic. See the recent survey papers by Yap and Dub e [49, 50]. Exact arithmetic is offered by the geometric software packages LEDA [13, 33] and CGAL [23, 38] Recently, Silva et al. 45] used an approach similar in spirit to the one described in this paper for ensuring reliability. While not addressing the robustness issue directly, they also describe how a ....

C.K. Yap. Robust Geometric Computation. In J.E. Goodman and J. O'Rourke, editors, CRC Handbook of Discrete and Computational Geometry, pages 653--668. CRC Press, 1997. ISBN ISBN 0-8493-8524-5.

....Two issues that are often ignored in the theoretical approach turn out to be critical in practice: Degeneracies and numerical precision. These issues are collectively referred to as robustness and they have been the topic of extensive research. Surveys on the topic can be found in [24] 27] [36], also several brief state of the art summaries on the topic are collected in [25] In theory degeneracies are often handled by assuming general position, namely assuming that degeneracies do not occur. The general position assumption had contributed significantly to the advancement of geometric ....

C. K. Yap. Robust geometric computation. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press LLC, 1997.

....[13] on the exact evaluation of integer determinants, Bronnimann et al. 12] on exact geometric predicates with single precision arithmetic, and Shewchuk s design and implementation [42] of four predicates based on adaptive floating point arithmetic. See the recent survey papers by Yap and Dub e [49, 48]. Exact arithmetic is offered by the geometric software packages LEDA [14, 32] and CGAL [23, 37] Recently, Silva et al. 44] have used an approach similar in spirit to the one described in this paper for ensuring reliability. While not addressing the robustness issue directly, they also describe ....

C.K. Yap. Robust Geometric Computation. In J.E. Goodman and J. O'Rourke, editors, CRC Handbook of Discrete and Computational Geometry, pages 653--668. CRC Press, 1997. ISBN ISBN 0-8493-8524-5.

....resolve a data induced degeneracy by computing a nondegenerate output that can be realized by an arbitrarily small perturbation of the input. A number of general approaches, based on symbolic perturbation schemes or other symmetry breaking techniques have been developed [10, 11, 16, 26, 27] see [28] for a recent survey. All of these techniques involve some computational overhead, often a substantial amount. In this paper, we consider the special case of two dimensional Delaunay triangulations. It is well known that not all possible triangulations have combinatorially equivalent realizations ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, CRC Handbook in Discrete and Computational Geometry, chapter 35. CRC Press, Boca Raton, FL, 1997.

.... intrinsically more ecient than previous general approaches (e.g. Wu s or Gr obner bases) 3) Our prover is implemented using the Core library [15, 13, 19] This is an unexpected application of our library, which was designed as a general C package to support the Exact Geometric Computation [26, 25] approach to robust algorithms. Preliminary experimental results are quite promising. We expect further improvements by ne tuning our library for this speci c application. Our prover is currently distributed with version 1.3 of the Core library (Aug. 15, 2000) and available from ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, 1997.

....during a program execution. The numerical non robustness problem has received much attention in the computational geometry community in the last 15 years ( 22, 38, 81, 49, 16, 23] The next section will briefly review some approaches to non robustness, we refer the reader to the current surveys ([86, 68, 64]) for more details. In [86] robustness literature was classified along two lines: the papers that aim to make fixed precision algorithms computation robust, and those that aim to make the exact computation approach e#cient. Call these the inexact and exact approaches, respectively, and the ....

....non robustness problem has received much attention in the computational geometry community in the last 15 years ( 22, 38, 81, 49, 16, 23] The next section will briefly review some approaches to non robustness, we refer the reader to the current surveys ( 86, 68, 64] for more details. In [86], robustness literature was classified along two lines: the papers that aim to make fixed precision algorithms computation robust, and those that aim to make the exact computation approach e#cient. Call these the inexact and exact approaches, respectively, and the corresponding algorithms Type I ....

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C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press LLC, 1997.

....The problem is that, when we incorporate non algebraic functions into our system, it is a major open problem whether a similar guarantee can be made. The concept of guaranteed accuracy is rooted in a general solution to the widespread problem of numerical non robustness in geometric algorithms [18,21]. This well known problem plagues many scientific and engineering computations. For instance, no computer aided design (CAD) software used in geometric modeling has any robustness guarantees they can be made to crash or give qualitatively wrong results with a suitable choice of inputs. ....

....we can make the comparison x : y without error. There is a fundamental gap between Levels II and III that may not be apparent: Level III is more than simply iterating a Level II computation with increasing precision. While we know how to provide Level III capability for all algebraic computation [21], it is an open question whether Level III is possible in the non algebraic case. We call this the fundamental problem of EGC. Let us isolate this critical issue precisely: let# be a set of real or complex functions or constants. In practice,# contains , Z, among other things. The set ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press LLC, Boca Raton, FL, 1997.

.... to be intrinsically more ecient than previous general approaches (e.g. Wu s or Gr obner bases) 3) Our prover is implemented using the Core library [13, 17] This is an unexpected application of our library, which was designed as a general C package to support the Exact Geometric Computation [23, 22] approach to robust algorithms. Preliminary experimental results are quite 122 promising. We expect further improvements by ne tuning our library for this speci c application. Our prover is currently distributed with version 1.3 of the Core library (Aug 15, 2000) and available from ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, 1997.

....problem. But it may be another problem when potential users need to (i) modify the technique for their particular requirements, or (ii) extend it to related problems. Robust solutions, especially those based 2 on xed precision geometry , are particularly resistant to (i) and (ii) See [33] for a survey of robustness literature. In this paper, we describe the Core Library (Core for short) a new C C library for robust numeric and geometric computation. Our library API ( application programmer interface ) model, as rst proposed in [32] is able to deliver powerful robustness ....

C. K. Yap. Robust geometric computation. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, 1997.

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Yap, C. K., Robust geometric computation, in CRC Handbook on Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke (eds.), CRC Press, 1997, 653--668.

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C. K. Yap. Robust geometric computation. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653-668. CRC Press LLC, Boca Raton, FL, 1997. 3

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C. K. Yap. Robust geometric computation. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653--668. CRC Press LLC, Boca Raton, FL, 1997. 10

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C. K. Yap, Robust geometric computations, in: J. E. Goodman and J. O'Rourke, ed., Handbook of Discrete and Computational Geometry (CRC Press LLC, Boca Raton, FL, 1997) 653--668. 21

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C. K. Yap. Robust geometric computation. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 35, pages 653668. CRC Press LLC, Boca Raton, FL, 1997.

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