Book ChapterDOI
Packing Convex Polygons into Rectangular Boxes
Helmut Alt,Ferran Hurtado +1 more
- pp 67-80
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TLDR
Efficient algorithms are given for disjoint packing of two polygons with a runtime close to linear for tanslations and 0(03) for geneal isometries.Abstract:
We consider the problem of packing several convex polygons into minimum size rectangles. For this purpose the polygons may be moved either by translations only, or by combinations of translations and rotations. We investigate both cases, that the polygons may overlap when being packed or that they must be disjoint. The size of a rectangle to be minimized can either be its area or its perimeter. In the case of overlapping packing very efficient algorithms whose runtime is close to linear or 0(n log n) can be found even for an arbitrary number of polygons. Disjoint optimal packing is known to be NP-hard for arbitrary numbers of polygons. Here, efficient algorithms are given for disjoint packing of two polygons with a runtime close to linear for tanslations and 0(03) for geneal isometries.read more
Citations
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Journal ArticleDOI
A simple method for fitting of bounding rectangle to closed regions
Debasis Chaudhuri,Ashok Samal +1 more
TL;DR: A new approach for fitting of a bounding rectangle to closed regions is introduced based on simple coordinate geometry and uses the boundary points of regions to determine the directions of major and minor axes of the object.
Journal ArticleDOI
Aligning Two Convex Figures to Minimize Area or Perimeter
Hee-Kap Ahn,Otfried Cheong +1 more
TL;DR: Given two compact convex sets P and Q in the plane, the problem of finding a placement ϕP of P that minimizes the convex hull of Q is considered and exact near-linear time algorithms are achieved for all versions of the problem.
Journal ArticleDOI
Sparsest packing of two-dimensional objects
Tatiana E. Romanova,Alexander Pankratov,Igor Litvinchev,Sergiy Plankovskyy,Yevgen Tsegelnyk,Olga Shypul +5 more
TL;DR: The sparsest packing is aimed to place the objects in the container as distant as possible and the minimal Euclidean distance between the objects and the boundary of the container is maximised.
Journal Article
Computational Aspects of Packing Problems
TL;DR: A survey on results achieved about computability and complexity of packing problems, about approximation algorithms, and about very natural packing problems whose computational complexity is unknown.
Proceedings Article
Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons.
TL;DR: This work investigates the problem of finding a minimum-area container for the disjoint packing of a set of convex polygons by translations and develops efficient constant factor approximation algorithms for both optimization problems which are NP-hard.
References
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Book
Davenport-Schinzel sequences and their geometric applications
Micha Sharir,Pankaj K. Agarwal +1 more
TL;DR: A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Solving geometric problems with the rotating calipers
TL;DR: This paper shows that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once, and that this simple idea can be generalized in two ways.
Journal ArticleDOI
Determining the minimum-area encasing rectangle for an arbitrary closed curve
H. Freeman,R. Shapira +1 more
TL;DR: The method is of interest in certain packing and optimum layout problems because it consists of first determining the minimal-perimeter convex polygon that encloses the given curve and then selecting the rectangle of minimum area capable of containing this polygon.
Journal ArticleDOI
Finding minimal enclosing boxes
TL;DR: It is proven that at least two adjacent box sides are flush with edges of the hull of the convex hull, and this characterization enables the anO(n3) algorithm to find all minimal boxes for a set ofn points.
Journal ArticleDOI
An optimal algorithm for finding minimal enclosing triangles
TL;DR: Klee and Laskowski's O ( n log 2 n ) algorithm for finding all minimal area triangles enclosing a given convex polygon of n vertices is improved to Θ ( n), which is shown to be optimal both forFinding all minima and for finding just one.
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