## Extremal Polygon Containment Problems (1991)

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### BibTeX

@MISC{Toledo91extremalpolygon,

author = {Sivan Toledo},

title = {Extremal Polygon Containment Problems},

year = {1991}

}

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### Abstract

Given a convex polygonal object P and an environment consisting of polygonal obstacles, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo [Me], and the fixed size polygon placement algorithms developed by Leven and Sharir [LS, LS1], to obtain an algorithm that runs in time O(k 2 n# 4 (kn) log 3 (kn) log log(kn)). We also present several other e#cient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon. 1 Introduction Let P be a convex polygon having k vertices and edges, and let Q be a closed two dimensional space bounded by a collection of polygonal obstacles (the "environment") having altogether n...

### Citations

690 | Algorithms in Combinatorial Geometry - Edelsbrunner - 1987 |

284 |
Parallel merge sort
- Cole
- 1988
(Show Context)
Citation Context ...) per pair, or O(n 2 ) overall. The parallel version works by sorting all the endpoints of all free intervals for each vertexvertex contact. This takes O(log n) time and uses O(n) processors per pair =-=[Co], or-=- O(log n) time and O(n 2 ) processors overall. Using Cole’s improvement to Megiddo’s technique [Co1, Me] we can save a logarithmic factor in the running time and obtain: Theorem 4 Given a triangle... |

200 | Linear-time algorithms for linear programming in r 3 and related problems - Megiddo - 1983 |

198 |
On the `Piano Movers' Problem. II. General Techniques for Computing Topological Properties of Real Algebraic Manifolds
- Schwartz, Sharir
- 1983
(Show Context)
Citation Context ...t is to verify that comparisons involve only evaluations of signs of low degree polynomials in the unspecified δ. Indeed, a free placement of P in Q has to satisfy a set of algebraic constraints (see=-= [SS]-=-). In our case these constraints are mainly algebraic inequalities that describe the disjointness of P and Q. Computing a critical triple contact amounts to setting three inequalities as tight constra... |

194 | Linear programming in linear time when the dimension is fixed - Megiddo - 1984 |

158 |
Optimal point location in a monotone subdivision
- Edelsbrunner, Guibas, et al.
- 1986
(Show Context)
Citation Context ...Computing the normal diagram can be done in linear time, given a reasonable representation of the polyhedra (e.g. a quad-edge structure), preprocessing can be done in O(k) time using the technique of =-=[EGS]-=-, and the point location queries can be all done in O(n log k) time. Thus the overall running time of the algorithm is O(k + n log k). It is obvious that the technique can be applied to fixed-size que... |

105 | Slowing down sorting networks to obtain faster sorting algorithms - Cole - 1987 |

105 | Nonlinearity of Davenport-Schinzel sequences and of a generalized path compressioa scheme, Combinatorica 6 - Hart, Sharir - 1986 |

105 |
Parallelism in comparison problems
- Valiant
- 1975
(Show Context)
Citation Context ...sume that we have both an efficient sequential algorithm As for solving P(I, δ) at any given δ, and a parallel algorithm Ap, assumed to run in Valiant’s comparison-based model of parallel computat=-=ion [Va]-=-. We will denote the running time of As by Ts, the running time of Ap by Tp, and the number of processors it uses by P . Assume moreover that the flow of execution of Ap depends only on comparisons, e... |

95 |
A Kinetic Framework for Computational Geometry
- Guibas, Ramshaw, et al.
- 1983
(Show Context)
Citation Context ...ixed and P as movable, and we will use the vertex p1 as a reference point for P . Proposition 2.1 ([Ch]) If P and Q are convex, then C(P, Q) is a convex polygon with at most n edges. Proposition 2.2 (=-=[GRS]-=-) If P and Q are convex, then O(P, Q) is a convex polygon with at most n + k edges. 2.1 Computation of C(P, Q) We assume that both P and Q are convex. The procedure given below for the computation of ... |

89 | Applications of parametric searching in geometric optimization
- Agarwal, Sharir, et al.
- 1994
(Show Context)
Citation Context ...ves upon results obtained in previous papers that studied related problems. Megiddo’s technique can be applied to many other extremal containment problems. For example, Megiddo’s technique is used=-= in [AST]-=- to solve the problems of finding the largest stick (line segment) that can be placed in a simple polygon, and finding a placement for a stick that minimizes the maximum distance from the stick to a g... |

79 | P.: Sharp upper and lower bounds on the length of general DavenportSchinzel sequences - Agarwal, Sharir, et al. - 1989 |

61 |
Parallel computational geometry
- Aggarwal, Chazelle, et al.
- 1988
(Show Context)
Citation Context ...of the ti’s will be however functions of the expansion ratio of P . Step 3 is performed in parallel using the parallel algorithm for computing the convex hull of a plane point set, by Aggarwal et al=-=. [ACGOY]-=-, that works in O(log n) time and uses O(n) processors. We now combine the sequential and parallel algorithms to obtain an algorithm that computes the largest homothetic copy of P that can be placed i... |

38 |
The polygon containment problem
- Chazelle
- 1983
(Show Context)
Citation Context ...d-size polygon containment problem, in which (the convex) P is only allowed to translate and rotate and we wish to determine whether there is any placement of a copy of P inside Q [Ch, AB1]. Chazelle =-=[Ch] s-=-tudies the problem for the case where P and Q are arbitrary simple polygons and presents a naive algorithm that takes time O(k 3 n 3 (k + n) log(k + n)). A more restricted ∗ A preliminary version of... |

37 |
Planning a Purely Translational Motion for a Convex Object
- Leven, Sharir
- 1987
(Show Context)
Citation Context ... length of the list (in one pass over the list), and the same time bound applies to the pruning of the redundant nodes. The merging step can be done in time proportional to the length of Γ, which, by=-= [LS1] and-=- the comments in Section 2, is O(λ4(kn)). Hence the calculation of the lower envelope 12sΨL;O takes O(λ4(kn) log kn) time, so all these envelopes can be computed in overall time O(knλ4(kn) log kn)... |

33 |
Polygon placement under translation and rotation
- Avnaim, Boissonnat
- 1988
(Show Context)
Citation Context ...lle [Ch], who solves this case in time O(kn 2 ). Chazelle gives a simple solution to an even more restricted version in which P is a triangle; this version runs in time O(n 2 ). Avnaim and Boissonnat =-=[AB1]-=- present an algorithm for the case where both P and Q are non-convex, possibly non-connected polygons, which runs in time O(k 3 n 3 log(kn)). In another paper Avnaim and Boissonnat [AB] investigate th... |

24 |
An efficient motion planning algorithm for a convex rigid polygonal object in 2-dimensional polygonal space
- Kedem, Sharir
- 1990
(Show Context)
Citation Context ...efinitions and results from [LS], and note that the combinatorial bound derived in that paper can be somewhat improved. In section 6 we describe a variant of the fixed size containment algorithm from =-=[KS]-=-, which we use as a decision procedure, to decide whether a copy of P having some fixed expansion ratio can be placed in Q. In section 7 we give a parallel version of the fixed size containment algori... |

21 |
On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space, Discrete Comput
- Leven, Sharir
- 1987
(Show Context)
Citation Context ...ngle in a convex polygon. This can be regarded as a warm-up exercise that sheds some light on the general and more complex algorithm. In section 5 we state some neccesary definitions and results from =-=[LS]-=-, and note that the combinatorial bound derived in that paper can be somewhat improved. In section 6 we describe a variant of the fixed size containment algorithm from [KS], which we use as a decision... |

19 |
Reporting and counting segment intersections
- Chazelle
- 1986
(Show Context)
Citation Context ...r the second situation arises, we preprocess Q for segment intersection queries, that is, given a query segment, determine quickly whether it intersects an edge of Q. For this we use the technique of =-=[Ch1]-=-. Again, this can be implemented with O(n 2 ) storage, O(n 2 ) preprocessing, and can answer a segment intersection query in O(log n) time. For each critical placement of P , we query this structure w... |

17 |
K.: Placing the largest similar copy of a convex polygon among polygonal obstacles
- Chew, Kedem
- 1989
(Show Context)
Citation Context ...Q is an arbitrary polygonal environment, this problem is solved in time O(kn log(kn)) by constructing a generalized Voronoi diagram of Q under a convex distance function induced by P . Chew and Kedem =-=[CK]-=- follow a related approach to solve a more difficult variant of the problem, in which P is also allowed to rotate, which is also the main problem studied in this paper. Instead of a Voronoi diagram, t... |

10 |
Fast algorithms for polygon containment
- FORTUNE
- 1985
(Show Context)
Citation Context ... or three not necessarily convex polygons in a closed polygonal environment. For this problem they allow translations only. Extremal polygon containment problems were also previously studied. Fortune =-=[Fo]-=-, and Leven and Sharir [LS1] consider the following problem: find the largest homothetic copy of P inside Q. In other words, translation and scaling of P are allowed, but rotation is not. When P is co... |

6 | Point retrieval for polygons - Paterson, Yao - 1986 |

4 | High-clearance motion planning for a convex polygon among polygonal obstacles - Chew, Kedem - 1990 |

4 |
Applying parallel computation in the design of serial algorithms
- Megiddo
- 1983
(Show Context)
Citation Context ...obstacles, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo =-=[Me],-=- and the fixed size polygon placement algorithms developed by Leven and Sharir [LS, LS1], to obtain an algorithm that runs in time O(k 2 nλ4(kn) log 3 (kn) log log(kn)). We also present several other... |

3 |
The polygon containment problem: 1. Simultaneous containment under translation
- Avnaim, Boissonnat
- 1987
(Show Context)
Citation Context ...nd Boissonnat [AB1] present an algorithm for the case where both P and Q are non-convex, possibly non-connected polygons, which runs in time O(k 3 n 3 log(kn)). In another paper Avnaim and Boissonnat =-=[AB]-=- investigate the problem of simultaneous placement of two or three not necessarily convex polygons in a closed polygonal environment. For this problem they allow translations only. Extremal polygon co... |