## The Graham Scan Triangulates Simple Polygons (1991)

Venue: | Pattern Recogn. Lett |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Kong91thegraham,

author = {Xianshu Kong and Hazel Everett and Godfried Toussaint},

title = {The Graham Scan Triangulates Simple Polygons},

journal = {Pattern Recogn. Lett},

year = {1991},

volume = {11},

pages = {11--713}

}

### OpenURL

### Abstract

The Graham scan is a fundamental backtracking technique in computational geometry which was originally designed to compute the convex hull of a set of points in the plane and has since found application in several different contexts. In this note we show how to use the Graham scan to triangulate a simple polygon. The resulting algorithm triangulates an n vertex polygon P in O(kn) time where k-1 is the number of concave vertices in P. Although the worst case running time of the algorithm is O(n 2 ), it is easy to implement and is therefore of practical interest. 1. Introduction A polygon P is a closed path of straight line segments. A polygon is represented by a sequence of vertices P = (p 0 ,p 1 ,...,p n-1 ) where p i has real-valued x,y-coordinates. We assume that no three vertices of P are collinear. The line segments (p i ,p i+1 ), 0 i n-1, (subscript arithmetic taken modulo n) are the edges of P. A polygon is simple if no two nonconsecutive edges intersect. A simple polygon part...

### Citations

263 |
An ecient algorithm for determining the convex hull of a planar set
- Graham
(Show Context)
Citation Context ...(n 2 ) if the prune-and-search algorithm in [EET] is used to find an ear in linear time. The Graham scan is an important technique in computational geometry which was independently proposed by Graham =-=[Gr]-=- to compute the convex hull of a sorted set of points and by Sklansky [Sk] to compute the convex hull of a simple polygon. Whereas the Sklansky scan fails for simple polygons [By] it succeeds for star... |

76 |
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- 1982
(Show Context)
Citation Context ... j of P is called a diagonal of P if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons =-=[Ch]-=- [CI] [FM] [GJPT] [HM] [KKT] [To] [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm ... |

69 |
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(Show Context)
Citation Context ...ed a diagonal of P if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] [CI] [FM] =-=[GJPT]-=- [HM] [KKT] [To] [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known is the O(n ... |

57 |
Triangulating simple polygons and equivalent problems
- Fournier, Montuno
- 1984
(Show Context)
Citation Context ... called a diagonal of P if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] [CI] =-=[FM]-=- [GJPT] [HM] [KKT] [To] [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known is t... |

44 | An Optimal Algorithm for Determining the Visibility of a Polygon from an Edge
- Avis, Toussaint
- 1981
(Show Context)
Citation Context ...n the Graham-Sklansky scan has found widespread application to other problems. For example, it has been used to determine in O(n) time whether a simple polygon is weakly visible from a specified edge =-=[AT]-=- and to triangulate in O(n) time a polygon known to be palm-shaped with respect to a point in the polygon [ET]. In this paper we show how to use the Graham scan to obtain an O(kn)-time implementation ... |

42 | Triangulation and shape-complexity
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- 1984
(Show Context)
Citation Context ... P is called a diagonal of P if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] =-=[CI]-=- [FM] [GJPT] [HM] [KKT] [To] [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known... |

41 |
Fast triangulation of a simple polygon
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- 1983
(Show Context)
Citation Context ...agonal of P if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] [CI] [FM] [GJPT] =-=[HM]-=- [KKT] [To] [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known is the O(n log l... |

36 | An O(n loglog n)-time algorithm for triangulating a simple polygon
- Tarjan, Wyk
- 1988
(Show Context)
Citation Context ...t lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] [CI] [FM] [GJPT] [HM] [KKT] [To] =-=[TV]-=-. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known is the O(n log log n)-time algor... |

34 |
Polygons Have Ears
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Citation Context ...1 the top of ear p i . We say that two ears p i and p j are non-overlapping if the interior of triangle (p i-1 ,p i ,p i+1 ) does not intersect the interior of triangle (p j-1 ,p j ,p j+1 ). Meisters =-=[Me]-=- has given an elegant inductive proof of the following theorem. - 2 - Two-Ears Theorem: Except for triangles every simple polygon has at least two non-overlapping ears. This theorem forms the basis of... |

29 |
Measuring concavity on a rectangular mosaic
- Sklansky
(Show Context)
Citation Context ...n linear time. The Graham scan is an important technique in computational geometry which was independently proposed by Graham [Gr] to compute the convex hull of a sorted set of points and by Sklansky =-=[Sk]-=- to compute the convex hull of a simple polygon. Whereas the Sklansky scan fails for simple polygons [By] it succeeds for star-shaped polygons, a fact upon which the correctness of the Graham scan rel... |

15 | Convex hull of a finite set of points in two dimensions - Bykat - 1978 |

8 |
On geodesic properties of polygons relevant to linear time triangulation, The Visual Computer 5
- ElGindy, Toussaint
- 1989
(Show Context)
Citation Context ...o determine in O(n) time whether a simple polygon is weakly visible from a specified edge [AT] and to triangulate in O(n) time a polygon known to be palm-shaped with respect to a point in the polygon =-=[ET]-=-. In this paper we show how to use the Graham scan to obtain an O(kn)-time implementation of the ear-cutting algorithm. Since k-1 is the number of concave vertices this algorithm can be as bad as O(n ... |

6 |
An output-complexity-sensitive polygon triangulation algorithm
- Toussaint
- 1988
(Show Context)
Citation Context ... if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] [CI] [FM] [GJPT] [HM] [KKT] =-=[To]-=- [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known is the O(n log log n)-time ... |

4 |
O(n log log n) polygon triangulation with simple data structures
- Kirkpatrick, Klawe, et al.
- 1990
(Show Context)
Citation Context ...l of P if it lies entirely inside P. A triangulation of a simple polygon consists of n-3 non-intersecting diagonals. Many algorithms exist for triangulating simple polygons [Ch] [CI] [FM] [GJPT] [HM] =-=[KKT]-=- [To] [TV]. These algorithms vary in their worst case time complexities, in the complexity of their descriptions and in the data structures they use. The fastest algorithm known is the O(n log log n)-... |

1 |
Slicing an Ear in Linear Time," internal memorandum
- ElGindy, Everett, et al.
- 1989
(Show Context)
Citation Context ... recursively triangulates the rest of the polygon. A brute-force implementation of this approach yields an O(n 3 )-time algorithm. This can be improved to O(n 2 ) if the prune-and-search algorithm in =-=[EET]-=- is used to find an ear in linear time. The Graham scan is an important technique in computational geometry which was independently proposed by Graham [Gr] to compute the convex hull of a sorted set o... |