Approximation of convex polygons

Alt, Helmut; Blömer, Johannes; Godau, Michael; Wagener, Hubert

doi:10.1007/bfb0032068

Cited by 35 publications

(20 citation statements)

References 4 publications

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“…A preliminary version of this result was presented in [4]. In [9] it was shown that it can be applied to convex surfaces in higher dimensions, as well.…”

Section: Closed Convex Curvesmentioning

confidence: 99%

Comparison of Distance Measures for Planar Curves

2003

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The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fréchet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called κ-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant κ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fréchet distance, as well.

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“…A preliminary version of this result was presented in [4]. In [9] it was shown that it can be applied to convex surfaces in higher dimensions, as well.…”

Section: Closed Convex Curvesmentioning

confidence: 99%

Comparison of Distance Measures for Planar Curves

2003

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“…When items are segments in the plane and only translations are allowed, a minimum-area convex suitcase can be found in O(n log n) time and a minimum-perimeter convex suitcase in O(n) time [7]. Somewhat related to our problem are works on shape matching, where maximizing the overlap (or equivalently, minimizing the symmetric difference) of two items under translations [9,14,33,51] or general rigid motions [12,13,27] has been considered.…”

Section: Related Workmentioning

confidence: 99%

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

et al. 2013

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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.

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“…Mount et al [10] studied the function mapping a translation vector to the area of overlap of a translated simple n-vertex polygon P with another simple m-vertex polygon Q, showing that it is continuous, piecewise polynomial of degree at most two, has O((nm) 2 ) pieces, and can be computed within the same time bound. No algorithm is known that computes the translation maximizing the area of overlap that does not essentially construct the whole function graph.…”

Section: Introductionmentioning

confidence: 99%

“…Alt et al [2] made some initial progress on a similar problem, showing, for instance, how to construct, for a convex polygon P , the axisparallel rectangle Q minimizing the symmetric difference of P and Q. In the case where P and Q are disjoint unions of n and m unit disks, de Berg et al [5] compute a (1 − ε)-approximation for the maximal area of overlap of P and Q under rigid motions in time O((n ) log m).…”

Section: Introductionmentioning

confidence: 99%

Maximizing the overlap of two planar convex sets under rigid motions

Ahn

Cheong

Park

et al. 2005

Proceedings of the Twenty-First Annual Symposium on Computational Geometry

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Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε > 0, we compute a rigid motion such that the area of overlap is at least 1 − ε times the maximum possible overlap. Our algorithm uses O(1/ε) extreme point and line intersection queries on P and Q, plus O((1/ε 2 ) log(1/ε)) running time. If only translations are allowed, the extra running time reduces to O((1/ε) log(1/ε)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/ε) log n + (1/ε 2 ) log(1/ε)) for rigid motions and O((1/ε) log n + (1/ε) log(1/ε)) for translations.

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Approximation of convex polygons

Cited by 35 publications

References 4 publications

Comparison of Distance Measures for Planar Curves

Comparison of Distance Measures for Planar Curves

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

Maximizing the overlap of two planar convex sets under rigid motions

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