scholarly journals The Complexity of Drawing a Graph in a Polygonal Region

Author(s):  
Anna Lubiw ◽  
Tillmann Miltzow ◽  
Debajyoti Mondal
Keyword(s):  
2002 ◽  
Vol 12 (04) ◽  
pp. 339-352 ◽  
Author(s):  
SANG-MIN PARK ◽  
JAE-HA LEE ◽  
KYUNG-YONG CHWA

We consider the problem of searching for mobile intruders in a polygonal region with one door by two guards. Given a simple polygon [Formula: see text] with a door d, which is called a room [Formula: see text], two guards start at d and walk along the boundary of [Formula: see text] to detect a mobile intruder with a laser beam between the two guards. During the walk, two guards are required to be mutually visible all the time and eventually meet at one point. We give a characterization of the class of rooms searchable by two guards and an O(n log n)-time algorithm to test if a given room admits a walk, where n is the number of the vertices in [Formula: see text].


2008 ◽  
Vol 23 (5) ◽  
pp. 728-739 ◽  
Author(s):  
Xue-Hou Tan ◽  
Bo Jiang
Keyword(s):  

2020 ◽  
Vol 10 (24) ◽  
pp. 8866
Author(s):  
Sangyoon Lee ◽  
Hyunki Hong ◽  
Changkyoung Eem

Deep learning has been utilized in end-to-end camera pose estimation. To improve the performance, we introduce a camera pose estimation method based on a 2D-3D matching scheme with two convolutional neural networks (CNNs). The scene is divided into voxels, whose size and number are computed according to the scene volume and the number of 3D points. We extract inlier points from the 3D point set in a voxel using random sample consensus (RANSAC)-based plane fitting to obtain a set of interest points consisting of a major plane. These points are subsequently reprojected onto the image using the ground truth camera pose, following which a polygonal region is identified in each voxel using the convex hull. We designed a training dataset for 2D–3D matching, consisting of inlier 3D points, correspondence across image pairs, and the voxel regions in the image. We trained the hierarchical learning structure with two CNNs on the dataset architecture to detect the voxel regions and obtain the location/description of the interest points. Following successful 2D–3D matching, the camera pose was estimated using n-point pose solver in RANSAC. The experiment results show that our method can estimate the camera pose more precisely than previous end-to-end estimators.


1985 ◽  
Vol 8 (2) ◽  
pp. 267-273 ◽  
Author(s):  
Marilyn Breen

LetSbe a polygonal region in the plane with edges parallel to the coordinate axes. If every5or fewer boundary points ofScan be partitioned into setsAandBso thatconv A⋃ conv B⫅S, thenSis a union of two convex sets, each a rectangle. The number5is best possible.Without suitable hypothesis on edges ofS, the theorem fails. Moreover, an example reveals that there is no finite Helly number which characterizes arbitrary unions of two convex sets, even for polygonal regions in the plane.


2001 ◽  
Vol 37 (19) ◽  
pp. 1164 ◽  
Author(s):  
V.P. Chumachenko

2020 ◽  
Vol 23 (3) ◽  
pp. 34-43
Author(s):  
Anatolii S. Il’inskii ◽  
Ivan S. Polyanskii ◽  
Dmitry E. Stepanov ◽  
Nikolay I. Kuznetsov

Annotation In this article, the use of the barycentric method is proposed for the numerical solution of problems of diffraction of electromagnetic waves on infinitely thin perfectly conducting screens of arbitrary shape. The numerical solution is formed in the projection formulation of the Galerkin method. The essence of the barycentric method is to form a global system of basic functions for opening the screen when determining the approximation of the desired function of the current plane on its surface. Basis functions are defined by Bernstein-type polynomials in terms of barycentric coordinates that are entered for opening the screen when it is represented as a closed simply connected polygonal region. The features of the algorithmic implementation of the barycentric method in solving diffraction problems on conducting thin screens are considered. The rate of convergence is estimated. Comparative results of calculations performed under equivalent conditions using the barycentric method and the RWG method are presented.


2019 ◽  
Vol 7 (1) ◽  
pp. 3-13 ◽  
Author(s):  
К. Панчук ◽  
K. Panchuk ◽  
Т. Мясоедова ◽  
T. Myasoedova ◽  
И. Крысова ◽  
...  

In this paper has been proposed a geometric model for forming problem of contour-parallel lines (equidistant lines) for a flat contour with an island, and has been obtained the problem’s analytical solution, which is relevant for computer-aided design of cutting tools processing pocket surfaces on CNC machines. The proposed geometric model is based on cyclograph mapping of space on a plane. Beyond the analytical solution the geometric model differs from the known algebraic models and their solutions for considered forming problem also by the fact that it allows obtain a more complete and evident representation on the relationship and interaction for all its geometric components at the stages of 3D computer visualization. A 3D geometric model based on a cyclograph mapping of space has been proposed for obtaining the families of equidistant lines for connected and multiply connected regions with closed contours taken as a basis for pocket surfaces modeling. An algorithm for the analytical solution of the problem related to equidistant families generation is getting from the geometric model. All stages of the analytical solution are accompanied by a figurative representation of geometric objects and their relations in the geometric model’s virtual electronic space. The proposed in this paper algorithm for the case of a doubly connected polygonal region can be used as a basis for generation of equidistant families for multiply connected polygonal regions. The presence of the analytical solution for the problem related to equidistant families generation simplifies greatly the automated calculation of the tool path and preparation of control programs for pocket surfaces manufacturing on CNC machines. Have been presented an example and algorithm providing support for working capacity of the proposed geometric model for considered forming problem.


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