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### Page 3506 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews
Summary: “We consider the following problem: Find a set of par- allel straight lines with equal spacing to hit all m grid points in a closed region bounded by a convex polygon P with n vertices such that  ...  We give a new proof that the interior of each simple polygon can be represented by a monotone Boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane  ...

### Minimum Convex Partitions and Maximum Empty Polytopes [article]

Adrian Dumitrescu, Sariel Har-Peled, Csaba D. Tóth
2014 arXiv   pre-print
Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers.  ...  Here we give a (1-ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1]^d.  ...  The authors thank Joe Mitchell for helpful discussions regarding the exact algorithms in Section 5, in particular for suggesting the reduction of the maximum-area-emptyconvex-body problem to the potato-peeling  ...

### Minimum Convex Partitions and Maximum Empty Polytopes [chapter]

Adrian Dumitrescu, Sariel Har-Peled, Csaba D. Tóth
2012 Lecture Notes in Computer Science
Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers.  ...  Here we give a (1 − ε)approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0, 1] d .  ...  The authors thank Joe Mitchell for helpful discussions regarding the exact algorithms in Section 5, in particular for suggesting the reduction of the maximum-area-emptyconvex-body problem to the potato-peeling  ...

### Recursive Calculation of Relative Convex Hulls [chapter]

Gisela Klette
2011 Lecture Notes in Computer Science
The relative convex hull of a simple polygon A, contained in a second simple polygon B, is known to be the minimum perimeter polygon (MPP).  ...  The paper recalls properties and algorithms related to the relative convex hull, and proposes a (recursive) algorithm for calculating the relative convex hull.  ...  The algorithm copies vertices of the convex hull of the inner polygon one by one until it finds a cavity.  ...

### Page 7102 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews
grid convex polygon for a convex region on the plane.  ...  The region bounded by this tightest boundary is called the maximum grid convex polygon (MGCP).  ...

### Digital Convexity and Cavity Trees [chapter]

Gisela Klette
2014 Lecture Notes in Computer Science
convex and concave parts of a boundary of a digital region.  ...  We review different approaches and we propose cavity trees for (a) analyzing the convexity of digital objects, (b) to decompose those objects into meaningful parts, and (c) to show an easy way to find  ...  Applying one of the definitions for digital convexity of regions, we can follow that the given digital region R contains at least one discrete point of the complement R of R and it is not convex.  ...

### Page 8108 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews
in the worst case for almost uniform grids of fixed di- mension d and O(logn) half-planes in the average for arbitrary planar grids.  ...  Given a d-variate convex function and an isothetic grid of size O(n“) in R“, which is supposed to be finite but not necessarily regular, we want to find the grid cell containing the minimum point.  ...

### Page 2105 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews
2105 68W The main idea is to use a dynamic programming algorithm based on finding optimal partitions for a number of “legal” polygons of varying size.  ...  In this paper we give a very simple algorithm to find a grid drawing of any given 4- connected plane graph G with four or more vertices on the outer face.  ...

### Convex Grid Drawings of Plane Graphs with Rectangular Contours [chapter]

Akira Kamada, Kazuyuki Miura, Takao Nishizeki
2006 Lecture Notes in Computer Science
A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an n × n grid if G is triconnected or the triconnected  ...  In a convex grid drawing, all vertices are put on grid points.  ...  This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of polynomial size.  ...

### Visibility, occlusion, and the aspect graph

Harry Plantinga, Charles R. Dyer
1990 International Journal of Computer Vision
In this paper we present tight bounds on the maximum size of the VSP and the aspect graph and give algorithms for their construction, first in the convex case and then in the general case.  ...  This enables us to find maximal regions of viewpoints of the same aspect.  ...  Canny [34] computes the region of a plane of viewpoints from which a polygon in R 3 is visible and gives an algorithm for constructing the "shadow" of a polygon relative to another polygon on the plane  ...

### Convex Grid Drawings of Plane Graphs with Rectangular Contours

Kazuyuki Miura, Akira Kamada, Takao Nishizeki
2008 Journal of Graph Algorithms and Applications
A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an n × n grid if G is triconnected or the triconnected  ...  In a convex grid drawing, all vertices are put on grid points.  ...  This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of polynomial size.  ...

### Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming [chapter]

Robin Deits, Russ Tedrake
2015 Springer Tracts in Advanced Robotics
program that finds a maximum-volume ellipsoid inside the polytope intersection of the obstacle-free half-spaces defined by those hyperplanes.  ...  The algorithm alternates between two convex optimizations: (1) a quadratic program that generates a set of hyperplanes to separate a convex region of space from the set of obstacles and (2) a semidefinite  ...  The authors also wish to thank the members of the Robot Locomotion Group at CSAIL for their advice and help.  ...

### On the number of sides necessary for polygonal approximation of black-and-white figures in a plane

James R. Ellis, Murray Eden
1976 Information and Control
A bound on the number of extreme points or sides necessary to approximate a convex planar figure by an enclosing polygon is described.  ...  This number is found to be proportional to the fourth root of the figure's area divided by the square of a maximum Euclidean distance approximation parameter.  ...  For the purpose of proof, we will inscribe a polygon in the convex set. Then, a central point may be picked for the purpose of forming disjoint sets covering the polygon.  ...

### Minimum vertex hulls for polyhedral domains [chapter]

Gautam Das, Deborah Joseph
1990 Lecture Notes in Computer Science
Joseph, Minimum vertex hulls for polyhedral domains, Theoretical Computer Science 103 (1992) 107-135: Given a collection of pairwise disjoint polygons on the plane, we wish to cover each polygon with an  ...  The paper also describes several approximations and exact algorithms for the problem.  ...  A convex polygon is one whose enclosed region is convex. A rectilinear polygon is one where each edge is either horizontal or vertical.  ...

### Minimum vertex hulls for polyhedral domains

Gautam Das, Deborah Joseph
1992 Theoretical Computer Science
Joseph, Minimum vertex hulls for polyhedral domains, Theoretical Computer Science 103 (1992) 107-135: Given a collection of pairwise disjoint polygons on the plane, we wish to cover each polygon with an  ...  The paper also describes several approximations and exact algorithms for the problem.  ...  A convex polygon is one whose enclosed region is convex. A rectilinear polygon is one where each edge is either horizontal or vertical.  ...
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