## FIST: Fast industrial-strength triangulation of polygons (1998)

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Venue: | Algorithmica |

Citations: | 11 - 2 self |

### BibTeX

@TECHREPORT{Held98fist:fast,

author = {Martin Held},

title = {FIST: Fast industrial-strength triangulation of polygons},

institution = {Algorithmica},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y

### Citations

636 | The LEDA Platform of Combinatorial and Geometric Computing - Mehlhorn, Näher - 1999 |

503 |
Computational Geometry in C
- O’ROURKE
- 1998
(Show Context)
Citation Context ...n) time spent on checking whether three subsequent vertices form an ear. A simple re-organization of the ear-finding process allows to check only O(n) ears, thus finding all ears in O(n 2 ) time; see =-=[32, 37]-=-. However, this complexity still is too high for practical applications. Whereas faces of a polyhedron typically have rather few vertices, applications in CAD/CAM and GIS may involve polygons with sev... |

403 | Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator
- Shewchuk
- 1996
(Show Context)
Citation Context ... we compared the cpu-time consumption of FIST and to those of other popular C codes: we compared it to Narkhede and Manocha's implementation [35, 36] of Seidel's algorithm [41], Shewchuk's "Trian=-=gle" [42, 44]-=-, Sloan's implementation [46] of an ear-clipping algorithm, and to Saade's implementation [40] of Toussaint's algorithm [48]. As witnessed by our tests, FIST is competitive with the other codes, and e... |

289 |
Triangulating a simple polygon in linear time
- Chazelle
- 1991
(Show Context)
Citation Context ...ries of improvements, the worst-case complexity of triangulating a polygon with n vertices has been brought down from O(n 2 ) to the optimal O(n), achieved by a seminal algorithm designed by Chazelle =-=[15]-=-. Algorithms with a slightly super-linear expected complexity also are known, cf. Seidel's [41] randomized algorithm. Virtually all published triangulation algorithms assume that the polygon is simple... |

276 | Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
- Edelsbrunner, Mucke
- 1990
(Show Context)
Citation Context ...not cope with data deficiencies and would yield judgments that (intuitively) were clearly incorrect for polygonal areas that had deficiencies. We emphasize, 5 For instance, see Edelsbrunner and Mucke =-=[21]. though, -=-that our approach to determining the orientations and the containment relation of polygonal loops is not guaranteed to make the "correct" judgment call for any form of twisted or self-inters... |

227 | Efficient collision detection using bounding volume hierarchies of k-DOPs
- Klosowski, Held, et al.
- 1998
(Show Context)
Citation Context ...l thousands of vertices. We try to reduce the practical complexity of finding one ear by performing only local searches. Bounding-volume trees, as successfully used in our work on collision detection =-=[27, 31]-=-, or, alternatively, regular grids are used for pruning the search while testing for earity 1 . While any form of geometric hashing does not reduce the worst-case complexity, it helps to reduce the pr... |

179 | Mesh generation and optimal triangulation
- Bern, Eppstein
- 1992
(Show Context)
Citation Context ...s hardware. Triangulating a polygon also is a fundamental operation in computational geometry, and it has received widespread interest over the last two decades; see Bern and Eppstein's recent survey =-=[10]-=-. Through a series of improvements, the worst-case complexity of triangulating a polygon with n vertices has been brought down from O(n 2 ) to the optimal O(n), achieved by a seminal algorithm designe... |

100 | A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications
- Seidel
- 1991
(Show Context)
Citation Context ...been brought down from O(n 2 ) to the optimal O(n), achieved by a seminal algorithm designed by Chazelle [15]. Algorithms with a slightly super-linear expected complexity also are known, cf. Seidel's =-=[41]-=- randomized algorithm. Virtually all published triangulation algorithms assume that the polygon is simple, i.e., that the vertices of the polygon are the only points of the plane that belong to two ed... |

94 | The exact computation paradigm
- Yap, Dubé
(Show Context)
Citation Context ...e floating-point arithmetic, cf. [43]. This code is also available publicly [42]. The issues of exact computation versus floating-point arithmetic are discussed in detail in a survey by Yap and Dub'e =-=[50]-=-. Recent work includes papers by Avnaim et al. [2] and Bronnimann and Yvinec [12] on the exact evaluation of integer determinants, Bronnimann et al. [11] on exact geometric predicates with single-prec... |

76 |
A theorem on polygon cutting with applications
- Chazelle
- 1982
(Show Context)
Citation Context ...erwards triangulated in linear time by repeatedly clipping off convex corners (i.e., ears) of the polygon. See also the work by Fournier and Montuno [24]. A completely different algorithm by Chazelle =-=[14]-=-, based on divide-and-conquer, also achieves an O(n log n) worst-case bound. The first triangulation algorithm that is sensitive to the shape of the polygon was given by Hertel and Mehlhorn [29]. Thei... |

72 | Robust geometric computation
- Yap
- 2004
(Show Context)
Citation Context ...redicates 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 ... |

66 |
Triangulating a simple polygon
- Garey, Johnson, et al.
- 1978
(Show Context)
Citation Context ...n) time in the worst case, earclipping algorithms need at least O(n 2 ) time in the worst case. Several O(n 2 ) algorithms based on ear clipping were proposed recently; see [22, 32, 37]. Garey et al. =-=[25]-=- were the first to publish an O(n log n) triangulation algorithm. They used a regularization scheme in order to partition a polygon in O(n log n) time into monotone sub-polygons. Monotone sub-polygons... |

65 | Piecewise-linear interpolation between polygonal slices
- Barequet, Sharir
- 1994
(Show Context)
Citation Context ...thm for determining a plane on which a projection of a given polygon is simple. Related to the triangulation of a 3D polygon is the linear interpolation of polygonal slices in 3D. Barequet and Sharir =-=[7]-=- use dynamic programming for determining a minimum-area triangulation between two polygonal slices. Their algorithm requires O(n 3 ) time and O(n 2 ) space. See also the work by Barequet et al. [6, 8,... |

54 |
Triangulating simple polygons and equivalent problems
- Fournier, Montuno
- 1984
(Show Context)
Citation Context ...onotone sub-polygons. Monotone sub-polygons are afterwards triangulated in linear time by repeatedly clipping off convex corners (i.e., ears) of the polygon. See also the work by Fournier and Montuno =-=[24]-=-. A completely different algorithm by Chazelle [14], based on divide-and-conquer, also achieves an O(n log n) worst-case bound. The first triangulation algorithm that is sensitive to the shape of the ... |

48 | Robust adaptive floating-point geometric predicates
- Shewchuk
- 1996
(Show Context)
Citation Context ... fine code that computes Delaunay triangulations and is able to perform guaranteed-quality meshing. "Triangle" is based on geometric predicates that rely on an adaptive floating-point arithm=-=etic, cf. [43]-=-. This code is also available publicly [42]. The issues of exact computation versus floating-point arithmetic are discussed in detail in a survey by Yap and Dub'e [50]. Recent work includes papers by ... |

42 | Triangulation and shape-complexity
- Chazelle, Incerpi
- 1984
(Show Context)
Citation Context ...r log r), where r denotes the number of reflex vertices of the polygon. See also the simple algorithm by Kong et al. [32] which runs in O(n \Delta (r + 1)) time. The algorithm by Chazelle and Incerpi =-=[16]-=- also takes advantage of the shape of the polygon. Their algorithm, which is based on trapezoidal decompositions, runs in time O(n log s), where s is the sinuosity of the polygon. (Roughly, the sinuos... |

41 | Computing exact geometric predicates using modular arithmetic with single precision
- Brönnimann, Emiris, et al.
- 1997
(Show Context)
Citation Context ...scussed in detail in a survey by Yap and Dub'e [50]. Recent work includes papers by Avnaim et al. [2] and Bronnimann and Yvinec [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 pap... |

40 | Evaluating signs of determinants using single-precision arithmetic, Algorithmica 17
- Avnaim, Boissonnat, et al.
- 1997
(Show Context)
Citation Context ... also available publicly [42]. The issues of exact computation versus floating-point arithmetic are discussed in detail in a survey by Yap and Dub'e [50]. Recent work includes papers by Avnaim et al. =-=[2]-=- and Bronnimann and Yvinec [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 implem... |

38 | SHARIR M.: Filling gaps in the boundary of a polyhedron
- BAREQUET
- 1995
(Show Context)
Citation Context ...ir [7] use dynamic programming for determining a minimum-area triangulation between two polygonal slices. Their algorithm requires O(n 3 ) time and O(n 2 ) space. See also the work by Barequet et al. =-=[6, 8, 9]-=-, and Choi and Park [17]. A planar polygon with i islands and a total of n edges can be triangulated in time O(n log n) by applying any of the standard techniques, such as plane sweep or regularizatio... |

38 |
Fast triangulation of simple polygons
- Hertel, Mehlhorn
- 1983
(Show Context)
Citation Context ...zelle [14], based on divide-and-conquer, also achieves an O(n log n) worst-case bound. The first triangulation algorithm that is sensitive to the shape of the polygon was given by Hertel and Mehlhorn =-=[29]-=-. Their algorithm achieves a complexity of O(n+r log r), where r denotes the number of reflex vertices of the polygon. See also the simple algorithm by Kong et al. [32] which runs in O(n \Delta (r + 1... |

38 | An O(n log log n)-time algorithm for triangulating a simple polygon
- Tarjan, Wyk
- 1988
(Show Context)
Citation Context ...tion of FIST and to those of other popular C codes: we compared it to Narkhede and Manocha's implementation [35, 36] of Seidel's algorithm [41], Shewchuk's "Triangle" [42, 44], Sloan's implementation =-=[46]-=- of an ear-clipping algorithm, and to Saade's implementation [40] of Toussaint's algorithm [48]. As witnessed by our tests, FIST is competitive with the other codes, and even outperformed them in most... |

35 | Exact geometric computation in LEDA
- Burnikel, Könnemann, et al.
- 1995
(Show Context)
Citation Context ...tation [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... |

35 |
Fast polygon triangulation based on Seidel's algorithm, in Graphics Gems
- Narkhede, Manocha
- 1995
(Show Context)
Citation Context ...s. Using synthetic test data generated by means of RPG [1], we compared the cpu-time consumption of FIST and to those of other popular C codes: we compared it to Narkhede and Manocha's implementation =-=[35, 36] of Seidel-=-'s algorithm [41], Shewchuk's "Triangle" [42, 44], Sloan's implementation [46] of an ear-clipping algorithm, and to Saade's implementation [40] of Toussaint's algorithm [48]. As witnessed by... |

35 | Designing the computational geometry algorithms library CGAL - Overmars - 1996 |

33 |
H.: Polygons have ears
- Meisters
- 1975
(Show Context)
Citation Context ...g open problem was to reduce this complexity by devising better algorithms. We note that all algorithms discussed below (except Shewchuk's "Triangle" [44]) are restricted to simple polygons.=-= Meisters [34]-=- proved that every simple polygon that is not a triangle has at least two non-overlapping ears. Since checking for earity takes O(n) time in the worst case, earclipping algorithms need at least O(n 2 ... |

29 | On triangulating three-dimensional polygons
- Barequet, Dickerson, et al.
- 1998
(Show Context)
Citation Context ...tions of visibility maps, runs in O(n) time. However, it seems to be too complicated to be implemented successfully. The situation is quite different when dealing with polygons in 3D. Barequet et al. =-=[5]-=- show that it is NP-complete to decide whether a 3D polygon has a triangulation which is not self-intersecting and which defines a simply-connected 2-manifold. Also, they give an O(n 4 ) algorithm for... |

28 | Efficient exact evaluation of signs of determinants
- Brönnimann, Emiris, et al.
- 1997
(Show Context)
Citation Context ... The issues of exact computation versus floating-point arithmetic are discussed in detail in a survey by Yap and Dub'e [50]. Recent work includes papers by Avnaim et al. [2] and Bronnimann and Yvinec =-=[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 predicate... |

27 |
Heuristics for the generation of random polygons
- Auer, Held
- 1996
(Show Context)
Citation Context ...led FIST as acronym for "Fast Industrial-Strength Triangulation ", has been tested extensively, and we present statistics on a series of test runs. Using synthetic test data generated by mea=-=ns of RPG [1], we compa-=-red the cpu-time consumption of FIST and to those of other popular C codes: we compared it to Narkhede and Manocha's implementation [35, 36] of Seidel's algorithm [41], Shewchuk's "Triangle"... |

24 |
A Fast Las Vegas Algorithm for Triangulating a Simple Polygon
- Clarkon, Tarjan, et al.
- 1989
(Show Context)
Citation Context ...gulated in linear time by using the algorithm by Fournier and Montuno [24]. An earlier randomized triangulation algorithm, also with O(n log ? n) expected complexity, was published by Clarkson et al. =-=[19]-=-. Independently and at the same time as Seidel, Clarkson et al. [18] also published a randomized parallel algorithm whose sequential version is very similar to Seidel's algorithm [41]. See also the wo... |

23 | Randomized parallel algorithms for trapezoidal diagrams
- Clarkson, Cole, et al.
- 1992
(Show Context)
Citation Context ...o [24]. An earlier randomized triangulation algorithm, also with O(n log ? n) expected complexity, was published by Clarkson et al. [19]. Independently and at the same time as Seidel, Clarkson et al. =-=[18]-=- also published a randomized parallel algorithm whose sequential version is very similar to Seidel's algorithm [41]. See also the work by Devillers [20]. Tarjan and Van Wyk [47] were the first to brea... |

22 | Automatic generation of triangular irregular networks using greedy cuts
- Silva, Mitchell, et al.
- 1995
(Show Context)
Citation Context ...oating-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 carefully tailored ear-cli... |

15 | Randomization yields simple O(n log ∗ n) algorithms for difficult Ω(n) problems
- Devillers
- 1992
(Show Context)
Citation Context ...and at the same time as Seidel, Clarkson et al. [18] also published a randomized parallel algorithm whose sequential version is very similar to Seidel's algorithm [41]. See also the work by Devillers =-=[20]-=-. Tarjan and Van Wyk [47] were the first to break the O(n log n) worst-case complexity. Their algorithm runs in O(n log log n) time. A simpler algorithm that achieves the same worst-case complexity wa... |

14 |
A heuristic triangulation algorithm for multiple planar contours using extended double-branching procedure
- Choi, Park
- 1994
(Show Context)
Citation Context ...g for determining a minimum-area triangulation between two polygonal slices. Their algorithm requires O(n 3 ) time and O(n 2 ) space. See also the work by Barequet et al. [6, 8, 9], and Choi and Park =-=[17]-=-. A planar polygon with i islands and a total of n edges can be triangulated in time O(n log n) by applying any of the standard techniques, such as plane sweep or regularization. The best bound known ... |

13 |
Triangulating disjoint Jordan chains
- Bar-Yehuda, Chazelle
- 1994
(Show Context)
Citation Context ...log n) by applying any of the standard techniques, such as plane sweep or regularization. The best bound known for the worst-case complexity is achieved by an algorithm due to Bar-Yehuda and Chazelle =-=[4], which ru-=-ns in (near-optimal) time O(n + i log 1+ffl i). Ronfard and Rossignac [39] describe a finite-state machine that is able to handle polygons with islands. Their algorithm, called "flooding", i... |

12 |
Polygon triangulation in O(n log log n) time with simple data structures. Discrete Comput
- Kirkpatrick, Klawe, et al.
- 1992
(Show Context)
Citation Context ... to break the O(n log n) worst-case complexity. Their algorithm runs in O(n log log n) time. A simpler algorithm that achieves the same worst-case complexity was later published by Kirkpatrick et al. =-=[30]-=-. Finally, Chazelle [15] ended the quest for a worst-case optimal triangulation algorithm. His ingenious construction, based on coarse approximations of visibility maps, runs in O(n) time. However, it... |

12 |
Triangulating Multiply-Connected Polygons: A Simple, Yet E cient Algorithm
- Ronfard, Rossignac
- 1994
(Show Context)
Citation Context ...ation. The best bound known for the worst-case complexity is achieved by an algorithm due to Bar-Yehuda and Chazelle [4], which runs in (near-optimal) time O(n + i log 1+ffl i). Ronfard and Rossignac =-=[39] describe a finite-s-=-tate machine that is able to handle polygons with islands. Their algorithm, called "flooding", is similar in spirit to sweep-line algorithms, but it sweeps only subparts ("gorges")... |

11 | Efficient triangulation of simple polygons
- Toussaint
- 1991
(Show Context)
Citation Context ...plementation [35, 36] of Seidel's algorithm [41], Shewchuk's "Triangle" [42, 44], Sloan's implementation [46] of an ear-clipping algorithm, and to Saade's implementation [40] of Toussaint's =-=algorithm [48]-=-. As witnessed by our tests, FIST is competitive with the other codes, and even outperformed them in most tests. This paper is accompanied by several color plates available on the WWW. Point your brow... |

10 |
Ecient and Reliable Triangulation of Polygons
- Held
- 1998
(Show Context)
Citation Context ...words: Polygon, triangulation, ear clipping, geometric hashing, reliability, robustness, experimental analysis. A preliminary version of this paper has appeared as an extended abstract at CGI'98; see =-=[26]-=-. y Email: held@cosy.sbg.ac.at. Most of this work was carried out while working at the Computational Geometry Lab of SUNY Stony Brook, and was supported by grants from Sun Microsystems, and by NSF Gra... |

4 | The Graham Scan Triangulates Simple Polygons
- Kong, Everett, et al.
- 1991
(Show Context)
Citation Context ...n) time spent on checking whether three subsequent vertices form an ear. A simple re-organization of the ear-finding process allows to check only O(n) ears, thus finding all ears in O(n 2 ) time; see =-=[32, 37]-=-. However, this complexity still is too high for practical applications. Whereas faces of a polyhedron typically have rather few vertices, applications in CAD/CAM and GIS may involve polygons with sev... |

3 | A Data Front-End for Layered Manufacturing
- Barequet, Kaplan
- 1997
(Show Context)
Citation Context ...ir [7] use dynamic programming for determining a minimum-area triangulation between two polygonal slices. Their algorithm requires O(n 3 ) time and O(n 2 ) space. See also the work by Barequet et al. =-=[6, 8, 9]-=-, and Choi and Park [17]. A planar polygon with i islands and a total of n edges can be triangulated in time O(n log n) by applying any of the standard techniques, such as plane sweep or regularizatio... |

3 |
et al. The cgal kernel: A basis for geometric computation
- Fabri
- 1996
(Show Context)
Citation Context ...r 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... |

2 |
Slicing an ear using prune-and-search
- ElGindy, Everett, et al.
- 1993
(Show Context)
Citation Context ...hecking for earity takes O(n) time in the worst case, earclipping algorithms need at least O(n 2 ) time in the worst case. Several O(n 2 ) algorithms based on ear clipping were proposed recently; see =-=[22, 32, 37]-=-. Garey et al. [25] were the first to publish an O(n log n) triangulation algorithm. They used a regularization scheme in order to partition a polygon in O(n log n) time into monotone sub-polygons. Mo... |

2 |
QuickCD: An Efficient Collision Detection System Using BV-Trees
- Held, Klosowski, et al.
- 1997
(Show Context)
Citation Context ...l thousands of vertices. We try to reduce the practical complexity of finding one ear by performing only local searches. Bounding-volume trees, as successfully used in our work on collision detection =-=[27, 31]-=-, or, alternatively, regular grids are used for pruning the search while testing for earity 1 . While any form of geometric hashing does not reduce the worst-case complexity, it helps to reduce the pr... |

2 |
et al. Exact geometric computation in leda
- Burnikel
- 1995
(Show Context)
Citation Context ...tation [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,... |

2 | RPG { Heuristics for the Generation of Random Polygons - Auer, Held - 1998 |

1 |
A Robust Method for Calculating the Simplicity and Orientation of Planar Polygons
- Balbes, Siegel
- 1991
(Show Context)
Citation Context ... algorithm first determines the orientation of every polygonal loop. Several approaches to determining the orientation of a polygon have been proposed, e.g., see the recent paper by Balbes and Siegel =-=[3]-=-. We settled on computing the signed area of a polygon for determining its orientation. The area of a polygon is computed by summing over the signed areas of the triangles \Delta(v 0 ; v i ; v i+1 ). ... |

1 | Optimizing a Corridor Between Two Polygons with an Application to Polyhedral Interpolation
- Barquet, Wolfers
- 1996
(Show Context)
Citation Context ...ir [7] use dynamic programming for determining a minimum-area triangulation between two polygonal slices. Their algorithm requires O(n 3 ) time and O(n 2 ) space. See also the work by Barequet et al. =-=[6, 8, 9]-=-, and Choi and Park [17]. A planar polygon with i islands and a total of n edges can be triangulated in time O(n log n) by applying any of the standard techniques, such as plane sweep or regularizatio... |

1 |
A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons
- Communication
- 1997
(Show Context)
Citation Context ...t to Narkhede and Manocha's implementation [35, 36] of Seidel's algorithm [41], Shewchuk's "Triangle" [42, 44], Sloan's implementation [46] of an ear-clipping algorithm, and to Saade's implementation =-=[40]-=- of Toussaint's algorithm [48]. As witnessed by our tests, FIST is competitive with the other codes, and even outperformed them in most tests. This paper is accompanied by several color plates availab... |

1 | QuickCD: An E cient Collision Detection System Using BV-Trees - Held, Klosowski, et al. - 1997 |