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## Solving geometric problems with the rotating calipers (1983)

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Citations: | 147 - 11 self |

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4844 |
Pattern classification and scene analysis
- Duda, Hart
- 1973
(Show Context)
Citation Context ...ax (P,Q), is defined as d max (P,Q) = {d(p i , q j )} i,j = 1, 2,..., n, where d(p i , q j ) is the euclidean distance between p i and q j . This distance measure has applications in cluster analysis =-=[5]-=-. A rather complicated O(n) algorithm for this problem appears in [6]. However, a very simple solution can be obtained by using a pair of calipers as in Figure 3. In Figure 3 the parallel lines of sup... |

534 | Computational Geometry.
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...rtices in standard form, i.e., the vertices are specified according to cartesian coordinates in a clockwise order and no three consecutive vertices are colinear. We assume the reader is familiar with =-=[1]-=-. In [1] Shamos presents a very simple algorithm for computing the diameter of P. The diameter is the greatest distance between parallel lines of support of P. A line L is a line of support of P if th... |

404 | An algorithm for planning collision-free paths among polyhedral obstacles.
- Lozano-Perez, Wesley
- 1979
(Show Context)
Citation Context ...noted as PsQ is the set consisting of all the elements obtained by adding every point in Q to every point in P. Vector sums of polygons and polyhedra have applications in collision avoidance problems =-=[8]-=-. The following theorems make the problem computable. Theorem 4.1: PsQ is a convex polygon. Theorem 4.2: PsQ has no more than 2n vertices. Theorem 4.3: The vertices of PsQ are vector sums of the verti... |

141 |
Convex hulls of finite sets of points in two and three dimensions,”
- Preparata, Hong
- 1977
(Show Context)
Citation Context ...the convex hull of a set of n points on the plane consists of sorting the points along the x axis and subsequently merging bigger and bigger convex polygons until one final convex polygon is obtained =-=[9]-=-. Performing the merge in linear time will guarantee an O(n log n) upper bound on the complexity of the entire process. Merging two convex polygons P, Q consists of essentially finding two pairs of ve... |

66 |
Determining the minimum-area encasing rectangle for an arbitrary closed curve,”
- Freeman, Shapira
- 1975
(Show Context)
Citation Context ...ween sets. 2. The Smallest-Area Enclosing Rectangle This problem has received attention recently in the image processing literature and has applications in certain packing and optimal layout problems =-=[2]-=- as well as automatic tariffing in goodstraffic [3]. Freeman and Shapira [2] prove the following crucial theorem for solving this problem Theorem 2.1: The rectangle of minimum area enclosing a convex ... |

52 |
Pattern recognition and geometrical complexity, in:
- Toussaint
- 1980
(Show Context)
Citation Context ...ctangle can be computed in constant time in this way resulting in a total running time of O(n). Another O(n) algorithm that implements this idea using a data structure known as a star is described in =-=[4]-=-. 3. The Maximum Distance Between Two Convex Polygons Let P = (p 1 , p 2 ,..., p n ) and Q = (q 1 , q 2 ,..., q n ) be two convex polygons. The maximum distance between P and Q, denoted by d max (P,Q)... |

36 |
On Translating a Set of Rectangles, in:
- Guibas, Yao
- 1983
(Show Context)
Citation Context ...ixed region. For two convex polygons the two CS lines partition the plane into the required visibility regions. 6.2 Collision avoidance Visibility and collision avoidance problems are closely related =-=[11]. Given tw-=-o convex polygons P and Q we may ask whether Q can be translated by an arbitrary amount in a specified direction without "colliding" with P. The CS lines provide an answer to this question. ... |

22 |
An on-line algorithm for fitting straight lines between data ranges,”
- O’Rourke
- 1981
(Show Context)
Citation Context ...without "colliding" with P. The CS lines provide an answer to this question. 6.3 Range fitting and linear separability Both of these problems involve finding a line that separates two convex=-= polygons [12]-=-. The critical support lines provide one solution to these problems. Consider Figure 6, where L(p i , q j ) and L(p i-2 ,q j-2 ) are the two CS lines. Denote their intersection by l*. We can choose as... |

11 | Efficient algorithms for computing the maximum distance between two finite planar sets
- Bhattacharya, Toussaint
- 1983
(Show Context)
Citation Context ... n, where d(p i , q j ) is the euclidean distance between p i and q j . This distance measure has applications in cluster analysis [5]. A rather complicated O(n) algorithm for this problem appears in =-=[6]-=-. However, a very simple solution can be obtained by using a pair of calipers as in Figure 3. In Figure 3 the parallel lines of support L s (q j ) and L s (p i ) have opposite directions and thus p i ... |

6 |
Pattern Synthesis
- Grenander
- 1976
(Show Context)
Citation Context ... i l*q j-2 . 6.4 The Grenander distance Given two disjoint convex polygons P and Q there are many ways of defining the distance between P and Q. One method already discussed is d max (P,Q). Grenander =-=[13]-=- uses a distance measure based on CS lines. Let LE(p i , p j ) denote the sum of the edge lengths of the polygonal chain p i , p i+1 ,..., p j-1 , p j and refer to Figure 6. The distance between P and... |

5 | A.: A simple O(n log n) algorithm for finding the maximum distance between two finite planar sets
- Toussaint, McAlear
- 1982
(Show Context)
Citation Context ...ameter problem of Shamos [1]. Note that d max (P,Q)sdiameter (PsQ) in general and thus we cannot use the diameter algorithm on PsQ to solve this problem. For further details the reader is referred to =-=[7]-=-. max i j , Fig. 3 q j q j+1 p i p i+1 p i+2 q i f j P Qs- 4. The Vector Sum of Two Convex Polygons Consider two convex polygons P and Q. Given a point r = (x r , y r ) e P and a point s = (x s , y s ... |

1 |
et al., "The smallest box around a package
- Groen
(Show Context)
Citation Context ... This problem has received attention recently in the image processing literature and has applications in certain packing and optimal layout problems [2] as well as automatic tariffing in goodstraffic =-=[3]-=-. Freeman and Shapira [2] prove the following crucial theorem for solving this problem Theorem 2.1: The rectangle of minimum area enclosing a convex polygon has a side collinear with one of the edges ... |

1 |
et al., "Graphics in flatland: a case study
- Edelsbrunner
- 1982
(Show Context)
Citation Context ...+1 q j-1 p i-1 Fig. 6 p i-2 q j-2s- A typical problem in two-dimensional graphics consists of computing all visibility lists of a set of objects for a tour or a path taken on the plane by an observer =-=[10]-=-. The first step of an algorithm for solving this task consists of partitioning the plane into regions R i such that the visibility list is the same for an observer stationed anywhere in some fixed re... |

1 |
et al., “The smallest box around a package
- Groen
(Show Context)
Citation Context ... This problem has received attention recently in the image processing literature and has applications in certain packing and optimal layout problems [2] as well as automatic tariffing in goodstraffic =-=[3]-=-. Freeman and Shapira [2] prove the following crucial theorem for solving this problem Theorem 2.1: The rectangle of minimum area enclosing a convex polygon has a side collinear with one of the edges ... |

1 |
et al., “Graphics in flatland: a case study
- Edelsbrunner
- 1982
(Show Context)
Citation Context ...y of problems. 6.1 Visibility - 6 -A typical problem in two-dimensional graphics consists of computing all visibility lists of a set of objects for a tour or a path taken on the plane by an observer =-=[10]-=-. The first step of an algorithm for solving this task consists of partitioning the plane into regions R i such that the visibility list is the same for an observer stationed anywhere in some fixed re... |