2,767 research outputs found
Implementation of linear minimum area enclosing traingle algorithm
This article has been made available through the Brunel Open Access Publishing Fund.An algorithm which computes the minimum area triangle enclosing a convex polygon in linear time already exists in the literature. The paper describing the algorithm also proves that the provided solution is optimal and a lower complexity sequential algorithm cannot exist. However, only a high-level description of the algorithm was provided, making the implementation difficult to reproduce. The present note aims to contribute to the field by providing a detailed description of the algorithm which is easy to implement and reproduce, and a benchmark comprising 10,000 variable sized, randomly generated convex polygons for illustrating the linearity of the algorithm
Maximal Area Triangles in a Convex Polygon
The widely known linear time algorithm for computing the maximum area
triangle in a convex polygon was found incorrect recently by Keikha et.
al.(arXiv:1705.11035). We present an alternative algorithm in this paper.
Comparing to the only previously known correct solution, ours is much simpler
and more efficient. More importantly, our new approach is powerful in solving
related problems
Approximating Smallest Containers for Packing Three-dimensional Convex Objects
We investigate the problem of computing a minimal-volume container for the
non-overlapping packing of a given set of three-dimensional convex objects.
Already the simplest versions of the problem are NP-hard so that we cannot
expect to find exact polynomial time algorithms. We give constant ratio
approximation algorithms for packing axis-parallel (rectangular) cuboids under
translation into an axis-parallel (rectangular) cuboid as container, for
cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary
convex container, and for packing convex polyhedra under rigid motions into an
axis-parallel cuboid or arbitrary convex container. This work gives the first
approximability results for the computation of minimal volume containers for
the objects described
A Generalization of the Convex Kakeya Problem
Given a set of line segments in the plane, not necessarily finite, what is a
convex region of smallest area that contains a translate of each input segment?
This question can be seen as a generalization of Kakeya's problem of finding a
convex region of smallest area such that a needle can be rotated through 360
degrees within this region. We show that there is always an optimal region that
is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute
such a triangle for a given set of n segments. We also show that, if the goal
is to minimize the perimeter of the region instead of its area, then placing
the segments with their midpoint at the origin and taking their convex hull
results in an optimal solution. Finally, we show that for any compact convex
figure G, the smallest enclosing disk of G is a smallest-perimeter region
containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure
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