Raab, S., Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes, in Proc. 15th ACM symposium on Computational Geometry, 1999, 163--172.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The use of exact computation turned out to be too slow for real time manipulation, so a finite precision method was needed. Controlled perturbation was devised to handle the robustness issues caused by the use of finite precision arithmetic, and to remove all the degeneracies. It was extended in [23], were it was applied to arrangements of polyhedral surfaces. Those arrangements require complex calculations in order to achieve a good perturbation bound. In [23] as in [14] the resolution bound (Section 4) is assumed to be given. The resolution bound is a key element in the scheme. In this ....

....handle the robustness issues caused by the use of finite precision arithmetic, and to remove all the degeneracies. It was extended in [23] were it was applied to arrangements of polyhedral surfaces. Those arrangements require complex calculations in order to achieve a good perturbation bound. In [23] (as in [14] the resolution bound (Section 4) is assumed to be given. The resolution bound is a key element in the scheme. In this work we describe a method for obtaining good resolution bounds, which we anticipate will lead to a better understanding of the method and will open the way to ....

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S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th ACM Sympos. Comput. Geom., pages 163--172, 1999.

....algorithm is very complex or not readily available. Other experiences suggest that property (iii) is the exception rather than the rule. In any case, users must weigh these considerations (cf. Sch94] A weaker form of the [BMS95] approach is illustrated by work of Halperin and co workers [HS98, Raa99] Again, the algorithm must explicitly detect the presence of degeneracies but now, we explicitly perturb the input to remove all degeneracies. Their problem may be framed as follows: given a sequence S = O 1 , On ) of geometric objects, let A i (i = 1, n) be the arrangement ....

S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th ACM Symposium on Computational Geometry, pages 163--172, 1999.

.... the sweep events in 30 types, and there are additional 18 types of events in the case of polyhedral surfaces [26] General position was achieved in the experiments in two ways: some of the sets were synthesized to guarantee general position, on others we ran a preprocessing perturbation step [24]. In order to verify that our programs run correctly we used two checking schemes. The rst scheme calculates the volume of each cell and checks that the sums of all the volumes is equal to the volume of the bounding simplex. The second scheme checks that the edges in the output graph G = U; E) ....

S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 163-172, 1999.

....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) Project under Contract No IST 2000 26473 (ECG E ective Computational Geometry for Curves and Surfaces) by The Israel Science Foundation founded by the Israel Academy of Sciences and ....

S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. M.Sc. thesis, Dept. Comput. Sci., Bar Ilan University, Ramat Gan, Israel, 1999. http://www.math.tau.ac.il/raab/master thesis.ps.

....cases. However since implicit integration methods [3, 5, 7] have been used for cloth simulations, the time step has become larger, sometimes as large as to meet the frame rate of 1 30 sec frame. Consequently this penetration problem is no longer negligible, so we use the swept volume approach [13, 8] which will be described in the next section. 2.1 Collision detection using swept volumes A swept volume is a volume made by two sets of positional entities of a face one at time t and one at time t #t. Connecting these old and new positions of all particles in a face gives us a volume. Any ....

S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. Anuual Symposium of Computational Geometry, Miami, FL, pages 163--172, June 1999.

....robust geometric algorithms. A variety of techniques have been proposed in recent years to overcome these diculties [10] 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 [9]. Snap rounding is a method of this type for converting an arrangement of segments into a low precision representation. Given a nite collection S of segments in the plane, the arrangement of S denoted A(S) is the subdivision of the plane into vertices, edges, and faces induced by S. A vertex of ....

S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. M.Sc. thesis, Dept. Comput. Sci., Bar Ilan University, Ramat Gan, Israel, 1999. http://www.math.tau.ac.il/raab/master thesis.ps.

....itself is irrelevant. Identi cation of all the degeneracies is only meaningful as a way to extract a consistent topology of the outer cell boundary but we will obtain the same goal in a much simpler way. We apply a controlled perturbation to the surfaces in P so that all degeneracies are removed [66], 67] One of our main goals here is to allow to manipulate the swept volume robustly with standard oating point arithmetic. We describe our scheme next. Since we aim to use standard oating point arithmetic, we are unable to tell for sure whether a degeneracy exists. We can only tell that a ....

....computation. Second, unlike various popular heuristic perturbation schemes (e.g. heuristic epsilons [69] our perturbation guarantees that the resulting collection of polyhedral surfaces is 10 D. Halperin Figure 8: Swept volume with many intersections (and potential degeneracies) in the middle [66] degeneracy free by carrying out a controlled incremental insertion process where we do not proceed to the next iteration before the arrangement induced by the subcollection of surfaces or subsurfaces of P we inserted so far is degeneracy free. Successful completion of the process is guaranteed ....

S. Raab. Controlled perturbation of arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th Annu. ACM Sympos. Comput. Geom., 1999.

....to the next iteration before the arrangement induced by the subcollection of surfaces or subsurfaces of P we inserted so far is degeneracy free. Successful completion of the process is guaranteed for values above a threshold which is determined by the analysis of the procedure. The thesis [65] contains a detailed report on all the degeneracies arising in arrangements of polyhedral surfaces and the swept volume computation, gives bounds on the perturbation radius as a function of and the input polyhedral surfaces, and describes the implementation of the scheme together with ....

....to achieve this goal, one of which is extending the scheme that we propose above, controlled perturbation, to other objects and to higher dimensions. Implementing the scheme is not dicult. However giving a theoretical guarantee for its viability for arrangements of complex objects is a dicult task [65]. As for motion planning, supporting only two and (currently partially) three dimensional arrangements Robust Geometric Computing in Motion 11 means we could solve robustly and accurately problems with a limited number of degrees of freedom. 6 Progress in the implementation of algorithms for ....

S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. M.Sc. thesis, Dept. Comput. Sci., Bar Ilan University, Ramat Gan, Israel, 1999.

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D. Halperin and S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. available from Halperin's home page; a preliminary version appeared in SoCG 1999, pages 163--172.

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S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes, in http://www.math.tau.ac.il/raab/master_thesis.ps. M.Sc. thesis, Dept. Comput. Sci., Bar Ilan University, Ramat Gan, Israel, 1999.

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S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 163172, 1999.

....is irrelevant. Identi cation of all the degeneracies is only meaningful as a way to extract a consistent topology of the outer cell boundary but we will obtain the same goal in a much simpler way. We apply a controlled perturbation to the surfaces in P so that all degeneracies are removed [66] [67]. One of our main goals here is to allow to manipulate the swept volume robustly with standard oating point arithmetic. We describe our scheme next. Since we aim to use standard oating point arithmetic, we are unable to tell for sure whether a degeneracy exists. We can only tell that a ....

.... testing) Most notably it uses the polyhedral surface package developed by Kettner [57] This work is an extension of an earlier work on arrangements of spheres as they arise in molecular modeling [50] In both cases (molecular modeling and swept volumes) we show by experiments [50] [67] that also on highly degenerate input (as depicted in Figure 8) the controlled perturbation scheme works eciently even when is fairly small. 5 Conclusions We described advancement in robust implementation of geometric algorithms. After reviewing the Cgal project and library we concentrated on ....

S. Raab and D. Halperin. Controlled perturbation of arrangements of polyhedral surfaces with application to swept volumes. Full version, manuscript, Tel Aviv University, 2000.

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Raab, S., Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes, in Proc. 15th ACM symposium on Computational Geometry, 1999, 163--172.

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S. Raab. Controlled perturbation of arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 163-172, 1999.

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S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th ACM Sympos. Comput. Geom., pages 163--172, 1999. 6

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S. Raab. Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 163-172, 1999.

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