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### Optimal placement of convex polygons to maximize point containment

Matthew Dickerson, Daniel Scharstein
1998 Computational geometry
Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P (allowing both translation and rotation) that contains the maximum number  ...  We then give a refinement that makes use of bucketing to improve the running time to O(nk2c2m 2 log(mk)), where c is the ratio of length to width of the polygon.  ...  Problem 1 (Optimal polygon placement). Given a convex polygon P and a planar point set S, find a rigid motion p that maximizes the number of points contained by p(P).  ...

### Offset-polygon annulus placement problems [chapter]

Gill Barequet, Amy J. Briggs, Matthew T. Dickerson, Michael T. Goodrich
1997 Lecture Notes in Computer Science
Optimization criteria include both maximizing the number of points contained in a fixed size annulus and minimizing the size of the annulus needed to contain all points.  ...  We address the following variants of the problem: placement of an annulus of a convex polygon as well as of a simple polygon; placement by translation only, or by translation and rotation; off-line and  ...  Given a set S of n points in the plane, a convex polygon P, and a distance 6, find a placement r of P that maximizes the number of points of S contained in the 3-annulus region of r(P).  ...

### Offset-polygon annulus placement problems

Gill Barequet, Amy J. Briggs, Matthew T. Dickerson, Michael T. Goodrich
1998 Computational geometry
Optimization criteria include both maximizing the number of points contained in a fixed size annulus and minimizing the size of the annulus needed to contain all points.  ...  We address the following variants of the problem: placement of an annulus of a convex polygon as well as of a simple polygon; placement by translation only, or by translation and rotation; off-line and  ...  Given a set S of n points in the plane, a convex polygon P, and a distance 6, find a placement r of P that maximizes the number of points of S contained in the 3-annulus region of r(P).  ...

### On Some Optimization Problems in Obnoxious Facility Location [chapter]

Zhongping Qin, Yinfeng Xu, Binhai Zhu
2000 Lecture Notes in Computer Science
We present a general method using Voronoi diagrams to approximately solve two such problems when the sites S's are points and weighted convex polygons (correspondingly, Vm's are points and weighted points  ...  Our algorithms run in polynomial time and approximate the optimal solutions of the above two problems by a factor of 2.  ...  Given a convex polygonal domain P and a set of point sites S in P , find the location of m undesirable or obnoxious facilities in P , which consist of a point set V m , so as to Maximize {d(V m , V m ∪  ...

### Translating a convex polygon to contain a maximum number of points

Gill Barequet, Matthew Dickerson, Petru Pau
1997 Computational geometry
Given a set S of n points in the plane and a convex polygon P with m vertices, we consider the problem of finding a translation of P that contains the maximum number of points in S.  ...  The algorithms also solve in the same running time the bichromatic variant of the problem, where we are given two point sets A and B and the goal is to maximize the number of points covered from A while  ...  Given a convex polygon P and a planar point set S, find a translation 7-that maximizes the number of points of S contained by T(P). We present two algorithms to solve Problem 1.  ...

### Optimizing a constrained convex polygonal annulus

Gill Barequet, Prosenjit Bose, Matthew T. Dickerson, Michael T. Goodrich
2005 Journal of Discrete Algorithms
Given a convex polygon P and a planar point set S, the goal is to find the thinnest annulus region of P containing S.  ...  We also provide solutions to a related known problem: finding the smallest homothetic copy of a polygon containing a set of points.  ...  Many thanks are also due to an anonymous referee for meticulously reading this manuscript and for many suggestions to improve it.  ...

### Computing the Maximum Overlap of Two Convex Polygons under Translations

M. de Berg, O. Cheong, O. Devillers, M. van Kreveld, M. Teillaud
1998 Theory of Computing Systems
Let P be a convex polygon in the plane with n vertices and let Q beaconvex polygon with m vertices.  ...  We prove that the maximum numberofcombinatorially distinct placements of Q with respect to P under translations is O(n 2 +m 2 + m i n (nm 2 +n 2 m)), and we g i v e an example showing that this bound is  ...  Thanks to H. Alt for helpful discussions about the centroid problem.  ...

### Computing the maximum overlap of two convex polygons under translations [chapter]

Mark de Berg, Olivier Devillers, Marc van Kreveld, Otfried Schwarzkopf, Monique Teillaud
1996 Lecture Notes in Computer Science
Let P be a convex polygon in the plane with n vertices and let Q beaconvex polygon with m vertices.  ...  We prove that the maximum numberofcombinatorially distinct placements of Q with respect to P under translations is O(n 2 +m 2 + m i n (nm 2 +n 2 m)), and we g i v e an example showing that this bound is  ...  Thanks to H. Alt for helpful discussions about the centroid problem.  ...

### The Discrete Voronoi Game in a Simple Polygon [chapter]

Aritra Banik, Sandip Das, Anil Maheshwari, Michiel Smid
2013 Lecture Notes in Computer Science
The objective of each player is to maximize the number of users they serve.  ...  Let P be a simple polygon with m vertices and let U be a set of n points in P . We consider the points of U to be "users". We consider a game with two players P1 and P2.  ...  Consider any optimal placement s for P 2 . Let γ be the cell that contains s. From the previous discussion, any point in γ acts as an optimal placement for . 3 .  ...

### Simulated Annealing with Adaptive Neighborhood Applied to the Placement over Containers with Fixed Dimensions

Thiago de Castro Martins, Marcos de Sales Guerra Tsuzuki
2008 IFAC Proceedings Volumes
This work deals with the problem of minimizing the waste of space that occurs on a rotational placement of a set of irregular bi-dimensional items inside a bi-dimensional container.  ...  The rotation applied and the translation of the polygon are continuous parameters, and the sequence of placement is represented as a set of discrete parameters.  ...  It can be defined as the problem of, given a container (a convex or concave polygon) and a set of n items (convex or concave polygons), determine the subset of items and transformations (translations and  ...

### A Flexible Framework for Covering and Partitioning Problems in Indoor Spaces

Sung-Hwan Kim, Ki-Joune Li, Hwan-Gue Cho
2020 ISPRS International Journal of Geo-Information
We apply it to particular applications, device placement and route planning problems, in order to give examples of the use of our framework in the perspective on how to design a constraint and how to use  ...  In this paper, we propose a multi-stage framework for indoor space partitioning, each stage of which can be flexibly adjusted according to target applications.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ...

### Optimized Packing Clusters of Objects in a Rectangular Container

T. Romanova, A. Pankratov, I. Litvinchev, Yu. Pankratova, I. Urniaieva
2019 Mathematical Problems in Engineering
A packing (layout) problem for a number of clusters (groups) composed of convex objects (e.g., circles, ellipses, or convex polygons) is considered.  ...  The objective of optimized packing is constructing a maximum sparse layout for clusters subject to nonoverlapping and containment conditions for clusters and objects.  ...  Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.  ...

### Improving the Reliability and Availability of Vehicular Communications Using Voronoi Diagram-Based Placement of Road Side Units

Prithviraj Patil, Aniruddha Gokhale
2012 2012 IEEE 31st Symposium on Reliable Distributed Systems
Effective placement of the RSUs is a key requirement in improving reliability of VANET services. This poster describes a novel Voronoi network-based algorithm for the effective placement of RSUs.  ...  The reliability of VANET-based services and applications that are based solely on vehicle-to-vehicle (V2V) communications, however, is hindered due primarily to limited and often fluctuating V2V communications  ...  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.  ...

### Optimal Camera Pose and Placement Configuration for Maximum Field-of-View Video Stitching

Alex Watras, Jae-Jun Kim, Hewei Liu, Yu Hu, Hongrui Jiang
2018 Sensors
An optimal camera placement problem is investigated. The objective is to maximize the area of the field of view (FoV) of a stitched video obtained by stitching video streams from an array of cameras.  ...  We present a method to find a convex polygon with maximum area from a given polygon.  ...  Conflicts of Interest: The authors declare no conflicts of interest.  ...

### Page 7003 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews
log Q) time.” 96k:68192 68U05 52B55 Dickerson, Matthew (1-MDLC-CS; Middlebury, VT); Scharstein, Daniel (1-CRNL-C; Ithaca, NY) Optimal placement of convex polygons to maximize point containment.  ...  with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation.  ...
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