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### Page 5146 of Mathematical Reviews Vol. , Issue 92i [page]

1992 Mathematical Reviews
into star-shaped polygons.  ...  We then use this algorithm to obtain an O(nlogn) ap- proximation algorithm for partitioning simple rectilinear polygons into star-shaped polygons with the size of the partition being at most six times  ...

### TERRAIN DECOMPOSITION AND LAYERED MANUFACTURING

SÁNDOR P. FEKETE, JOSEPH S. B. MITCHELL
2001 International journal of computational geometry and applications
We consider a problem that arises in generating three-dimensional models by methods of layered manufacturing: How does one decompose a given model P into a small number of sub-models each of which is a  ...  We also prove a two-dimensional version of this theorem, for the case in which P is a polygonal region with holes.  ...  Acknowledgments We are most grateful to Godfried Toussaint for useful discussions on these problems, help in formulating the problem, and organizing the 1993 Barbados Workshop on Geometric and Computational  ...

### Graph-Theoretic Solutions to Computational Geometry Problems [article]

David Eppstein
2009 arXiv   pre-print
We survey the art gallery problem, partition into rectangles, minimum-diameter clustering, rectilinear cartogram construction, mesh stripification, angle optimization in tilings, and metric embedding from  ...  Often, the efficiency of the algorithm depends on the special properties of the graph constructed in this way.  ...  Partition into rectangles Many geometric algorithms take as input a complicated polygonal domain and cover or partition it using simpler shapes  ; the partitions into triangles and quadrilaterals  ...

### Approximation Algorithms for Edge-Covering Problem [chapter]

2008 Communications in Computer and Information Science
Focus On Special Regions In A Polygon: One useful direction for approximation is to decompose the polygon into simple sub polygons (such as convex, star-shaped, monotone, etc.).  ...  A good candidate shape for such decomposition is the star-shaped polygon. We have attended some regions in a polygon where carry interesting features.  ...

### Visibilty: Finding the Staircase Kernel in Orthogonal Polygons

Stefan A. Pape, Tzvetalin S. Vassilev
2012 American Journal of Computational and Applied Mathematics
Based on this notion we can generalize the notion of star-shapedness.  ...  A polygon P is called star-shaped under staircase visibility, or simply s-star if and only if there is nonempty set of points S in the interior of P, such that any point of S sees any point of P via staircase  ...  Vassilev during the work on this paper.  ...

### Illumination of polygons by 45°-floodlights

Csaba D. Tóth
2003 Discrete Mathematics
We show also that every simple polygon with 2' + 2 vertices can be partitioned into ' quadrilaterals using at most ' − 1 Steiner points.  ...  What is the minimal number of oodlights that can illuminate the interior of any polygon with n vertices, provided that every oodlight has an ; ∈ (0 This question is answered in this paper for ∈ [45 • ;  ...  Every star ploygon can be partitioned into quadrilaterals by applying recursively the dissection described in the proof of Proposition 3 (see Fig. 8 ).  ...

### Efficient geometric algorithms on the EREW PRAM

D.Z. Chen
1995 IEEE Transactions on Parallel and Distributed Systems
set in the plane, triangulating monotone polygons and star-shaped polygons, computing the all dominating neighbors, etc.  ...  These problerns include: computing the convex hull oCa sorted point set in the plane, computing the convex hull oCa simple polygon, finding the kernel of a simple polygon, triangulating a sorted point  ...  Hence we optimally solve the problem of triangulating a star-shaped polygon.  ...

### Fitting Planar Graphs on Planar Maps

Md. Jawaherul Alam, Michael Kaufmann, Stephen G. Kobourov, Tamara Mchedlidze
2015 Journal of Graph Algorithms and Applications
We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.  ...  We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps.  ...  The cluster for the L-shaped polygon is partitioned by an (a b)-path into two convex polygons with the path as common boundaries.  ...

### Fitting Planar Graphs on Planar Maps [chapter]

Md. Jawaherul Alam, Michael Kaufmann, Stephen G. Kobourov, Tamara Mchedlidze
2014 Lecture Notes in Computer Science
We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.  ...  We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps.  ...  The cluster for the L-shaped polygon is partitioned by an (a b)-path into two convex polygons with the path as common boundaries.  ...

### Polygon Area Decomposition Using a Compactness Metric [article]

Mariusz Wzorek, Cyrille Berger, Patrick Doherty
2021 arXiv   pre-print
In this paper, we consider the problem of partitioning a polygon into a set of connected disjoint sub-polygons, each of which covers an area of a specific size.  ...  Results show that the proposed algorithm can efficiently divide polygon regions maximizing compactness of the resulting partitions, where the sub-polygon regions are on average up to 73% more compact in  ...  These include problems of polygon partitioning into trapezoids  ,  , rectangles  , convex polygons  ,  ,  or star-shaped polygons  .  ...

### Polygon Area Decomposition for Multiple-Robot Workspace Division

1998 International journal of computational geometry and applications
Polynomial-time algorithms have also been presented for decomposing polygons into trapezoids 2], convex polygons 6, 15, 17, 19, 21, 31] , star-shaped or monotone polygons 19], and rectangles 21, 23].  ...  This problem concerns dividing a given polygon P into n polygonal pieces, each of a speci ed area and each containing a certain point (site) on its boundary.  ...  Acknowledgements Thanks to Greg Sharp and Andrew Prock for helpful discussions and input on earlier drafts of this paper.  ...

### What makes a Tree a Straight Skeleton?

Oswin Aichholzer, Howard Cheng, Satyan L. Devadoss, Thomas Hackl, Stefan Huber, Brian Li, Andrej Risteski
2012 Canadian Conference on Computational Geometry
For small star graphs and caterpillars we show necessary and sufficient conditions for constructing P .  ...  We show that for given G such a polygon P might not exist, and if it exists it might not be unique. For the later case we give an example with exponentially many suitable polygons.  ...  Observe that the polygon P E(Sn) is star shaped and v i v i+1 (with v n+k := v 1+(k−1) mod n ) are its edges.  ...

### TURNING SHAPE DECISION PROBLEMS INTO MEASURES

RALPH R. MARTIN, PAUL L. ROSIN
2004 International journal of shape modeling
We give two general principles for constructing measures in this way, and show how they can be applied to construct various shape measures, including ones for convexity, circularity, ellipticity, triangularity  ...  , rectilinearity, rectangularity and symmetry in two dimensions, and 2.5D-ness, stability, and imperforateness in three dimensions.  ...  Acknowledgments We would like to thank the members of the Geometric Modelling Society for various helpful discussions, including suggesting some shape properties for measurement, and pointers to references  ...

### Guarding Path Polygons with Orthogonal Visibility [article]

Hamid Hoorfar, Alireza Bagheri
2017 arXiv   pre-print
A set of point guards in polygon P is named guard set (as denoted G ) if the union of visibility areas of these point guards be equal to polygon P i.e. every point in P be visible from at least one point  ...  For an orthogonal polygon, if dual graph of vertical decomposition is a path, it is named path polygon.  ...  . r-star is an orthogonal star-shaped polygon, and every r-star polygons are orthoconvex that will defined later.  ...

### Total Variation Denoising in \$l^1\$ Anisotropy

Michał Łasica, Salvador Moll, Piotr B. Mucha
2017 SIAM Journal of Imaging Sciences
We consider a naturally defined class of functions piecewise constant on rectangles (PCR).  ...  We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total  ...  It is also convenient to introduce the following notions of partitions of rectilinear polygons and signatures for their boundaries. Let Ω be a rectilinear polygon.  ...
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