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### Largest inscribed rectangles in convex polygons

Christian Knauer, Lena Schlipf, Jens M. Schmidt, Hans Raj Tiwary
2012 Journal of Discrete Algorithms
We consider approximation algorithms for the problem of computing an inscribed rectangle having largest area in a convex polygon on n vertices.  ...  If the order of the vertices of the polygon is given, we present a randomized algorithm that computes an inscribed rectangle with area at least (1 − ) times the optimum with probability t in time O ( 1  ...  Given a convex polygon P with n vertices, we want to compute a largest inscribed rectangle in P .  ...

### Largest Inscribed Rectangles in Geometric Convex Sets [article]

Mehdi Behroozi
2021 arXiv   pre-print
Finally, a parametrized optimization approach is developed to find the largest (axis-aligned) inscribed rectangles in two-dimensional space.  ...  To find the largest axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed.  ...  higher dimensions, we begin with the geometric properties of the traditional largest inscribed rectangles in a convex polygon in 2D.  ...

### Finding the Largest Volume Parallelepipedon of Arbitrary Orientation in a Solid

Ruben Molano, Daniel Caballero, Pablo G. Rodriguez, Maria Del Mar Avila, Juan P. Torres, Maria Luisa Duran, Jose Carlos Sancho, Andres Caro
2021 IEEE Access
There are no other approximation algorithms for finding the largest volume parallelepipedon of arbitrary orientation inscribed in a closed 3D contour with a computational cost better than the algorithm  ...  However, in many cases the objects are available as irregular shapes with many vertices, and in order to apply algorithms effectively, it is essential to compute the largest volume parallelepipedon contained  ...   ,  considered approximation algorithms to calculate the rectangle with the largest area of arbitrary orientation in a convex polygon with n vertices in O( 1 ε log 1 ε log n) for simple polygons  ...

### Approximation of convex figures by pairs of rectangles

Otfried Schwarzkopf, Ulrich Fuchs, Günter Rote, Emo Welzl
1998 Computational geometry
If the n vertices of a convex polygon C are given as a sorted array, such an approximating pair of rectangles can be computed in time O(log 2 n).  ...  We consider the problem of approximating a convex figure in the plane by a pair (r, R) of homothetic (that is, similar and parallel) rectangles with r C_ C C_ R.  ...  They compute the largest-area or the largest-perimeter rectangle with a given orientation contained in a polygon with n vertices in time O(log n).  ...

### Approximation of convex figures by pairs of rectangles [chapter]

Otfried Schwarzkopf, Ulrich Fuchs, Günter Rote, Emo Welzl
1990 Lecture Notes in Computer Science
If the n vertices of a convex polygon C are given as a sorted array, such an approximating pair of rectangles can be computed in time O(log 2 n).  ...  We consider the problem of approximating a convex gure in the plane by a pair (r; R ) of homothetic (that is, similar and parallel) rectangles with r C R.  ...  They compute the largest-area or the largest-perimeter rectangle with a given orientation contained in a polygon with n vertices in time O(logn).  ...

### Largest triangles in a polygon [article]

Seungjun Lee, Taekang Eom, Hee-Kap Ahn
2020 arXiv   pre-print
We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane.  ...  In the case with reorientations for convex polygons with n vertices, we also present (1-ε)-approximation algorithms.  ...   presented an algorithm of computing a largest axis-aligned rectangle that can be inscribed in a convex polygon. We follow their approach with some modification.  ...

### Maximum-Area Triangle in a Convex Polygon, Revisited [article]

Vahideh Keikha and Maarten Löffler and Ali Mohades and Jérôme Urhausen and Ivor van der Hoog
2017 arXiv   pre-print
We revisit the following problem: Given a convex polygon P, find the largest-area inscribed triangle.  ...  Also we show by example that the algorithm presented in 1979 by Dobkin and Snyder for finding the largest-area k-gon that is inscribed in a convex polygon fails to find the optimal solution for k=4.  ...  Acknowledgments The authors would like to thank everybody who has discussed this discovery and its implications with them over the past months, in particular Bahareh Banyassady, Ahmad Biniaz, Prosenjit  ...

### Linear programming in R3 and the skeleton and largest incircle of a convex polygon

M. Orlowski, M. Pachter
1987 Computers and Mathematics with Applications
Rodin Almtraet--In this paper the geometrical problem of constructing the largest circle inscribed in a (given) convex polygon is solved in 0(n) time.  ...  This problem is related to the construction of the skeleton of the polygon, which construction is shown to be accomplishable in 0(n log n) time.  ...  Smit of the Computer Science Division of NRIMS for presenting them with the problem of inscribing a circle in a polygon. The authors would also like to thank Dr D. H.  ...

### Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Hee-Kap Ahn, Peter Brass, Otfried Cheong, Hyeon-Suk Na, Chan-Su Shin, Antoine Vigneron
2006 Computational geometry
Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S that contains  ...  More precisely, for any ε > 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| > (1 − ε)|S| and we find an axially symmetric convex polygon Q containing C with area |Q | < (1 + ε)|S |.  ...  Acknowledgement We are grateful to an anonymous referee for a very careful proof-reading of our manuscript, and in particular suggesting the idea behind the linear-time algorithm of Lemma 15.  ...

### Finding the Maximum Area Parallelogram in a Convex Polygon

Kai Jin, Kevin Matulef
2011 Canadian Conference on Computational Geometry
We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an n-sided polygon in O(n 2 ) time.  ...  We also discuss applications of our result to the problem of computing the maximum area centrallysymmetric convex body (MAC) inside a given convex polygon, and to a "fault tolerant area maximization" problem  ...  The authors would also like to thank Xiaoming Sun, Tiancheng Lou, and Zhiyi Huang for taking part in fruitful discussions.  ...

### Convexity Problems on Meshes with Multiple Broadcasting

D. Bhagavathi, S. Olariu, J.L. Schwing, W. Shen, L. Wilson, J. Zhang
1995 Journal of Parallel and Distributed Computing
, and the largest-area inscribed triangle of a convex n-gon.  ...  Finally, we show that for two separable convex n-gons P and Q, the task of computing the minimum distance between P and Q can be performed in O(1) time on a mesh with multiple broadcasting of size n n.  ...  Their constructive criticism and comments have resulted in a much improved presentation. We wish to thank Professor Sahni for his timely and professional handling of our submission.  ...

### Planar Rectangular Sets and Steiner Symmetrization

Paul R. Scott
1998 Elemente der Mathematik
Paul Scott has worked in the Department of Pure Mathematics at the University Adelaide for the past 30 years. His research interests are in convex sets and the geometry of numbers.  ...  He is also interested in mathematics education, and has edited and typeset 'The Australian Mathematics Teacher' for the past seven years.  ...  Final Comment The class of rectangular sets appears naturally here in terms of successive orthogonal symmetrizations; to my knowledge, this class does not occur elsewhere in the literature.  ...

### A Novel Method for Recognizing Sketched Objects by Learning Their Geometrical Features

2020 jecet
This is currently important, particularly because of recent advances in human computer interactions via portable devices.  ...  In this paper, we present and explain a novel method for recognizing sketched object by learning a specific set of geometrical features, and creating a classification for these objects.  ...  polygons:  The largest area triangles that fits inside the convex hull.  The largest area enclosing rectangles.  ...

### Large Area Convex Holes in Random Point Sets

Octavio Arizmendi, Gelasio Salazar
2016 SIAM Journal on Discrete Mathematics
We also consider the problems of estimating the largest area convex hole, and the largest area of a convex polygonal hole with vertices in Kn.  ...  Let K, L be convex sets in the plane. For normalization purposes, suppose that the area of K is 1.  ...  ) convex hole; and let PolyMax(K n ) denote the random variable that measures the largest (in terms of area) convex polygon with vertices in K n .  ...

### Large area convex holes in random point sets [article]

Octavio Arizmendi, Gelasio Salazar
2015 arXiv   pre-print
We also consider the problems of estimating the largest area convex hole, and the largest area of a convex polygonal hole with vertices in K_n.  ...  Let K, L be convex sets in the plane. For normalization purposes, suppose that the area of K is 1.  ...  ) convex hole; and let PolyMax(K n ) denote the random variable that measures the largest (in terms of area) convex polygon with vertices in K n .  ...
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