Generalized self-approaching curves

Aichholzer, Oswin; Aurenhammer, Franz; Icking, Christian; Klein, Rolf; Langetepe, Elmar; Rote, Günter

doi:10.1016/s0166-218x(00)00233-x

Cited by 26 publications

(33 citation statements)

References 7 publications

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“…Examples for κ-straight curves are the curves with increasing chords of [13], where κ ≤ 2π/3, or the self-approaching curves of [2]. Given a polygonal curve P on n vertices and a parameter κ ≥ 1 one can decide in O(n log n) time whether P is κ-straight, see [1].…”

Section: κ-Straight 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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Section: κ-Straight Curvesmentioning

confidence: 99%

Comparison of Distance Measures for Planar Curves

2003

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“…The algorithm reports that a self-approaching path does not exist only in two cases: (1) if s lies in a dead region; (2) if an involute to a convex hull separates s from t. In both cases, any path from s to t will have to cross some dead region, and thus cannot be self-approaching.…”

Section: Casementioning

confidence: 99%

Self-approaching paths in simple polygons

Bose

Kostitsyna

Langerman

2020

Computational Geometry

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We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac| ≥ |bc|. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to use a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order.Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.• Given a polygon P , test if it is self-approaching, i.e., if there exists a self-approaching path between any two points in P .Related work. Self-approaching curves were first introduced in the context of online searching for a kernel of a polygon [13]. They were further studied in [14], where among other results, the authors prove that the length of any self-approaching path connecting two points is not greater than 5.3331 times the Euclidean distance between the points. An equivalent definition of a self-approaching path is that for every point on the path there has to be a 90 • angle containing the rest of the path. Aichholzer et al.[1] developed a generalization of self-approaching paths for an arbitrarily fixed angle α instead of 90 • . A relevant type of paths is the increasing chords paths [19], which are self-approaching in both directions. The nice properties of self-approaching and increasing chords paths and their potential to be applied in network routing were recognized by the graph drawing community. As a result, a number of papers appeared in the recent years on self-approaching and increasing chords graphs [2,9,18].This paper is organized in the following way. We introduce a few definitions and concepts in Section 2. In Section 3, we characterize a shortest self-approaching path between two points in a simple polygon. In Section 4 we present an algorithm to construct the shortest self-approaching path between two points if it exists, or to report that it does not exist, by assuming a model of computation in which we can solve certain transcendental equations. Finally, in Section 5 we present a linear-time algorithm to decide if a polygon is self-approaching, that is, if there is a self-approaching path between any two point of the polygon. In Section 6, we discuss some properties of reachable and reverse-reachable regions. PreliminariesFor two points p 1 and p 2 on a directed path π that starts in point s, we shall say that p 1 < π p 2 if p 1 lies between s and p 2 along π. For a directed path π and two...

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“…Sometimes the geometric properties of curves allow us to infer upper bounds on their detour [4,18,24], but these results do not lead to efficient computation of the detour of the curve.…”

Section: Introductionmentioning

confidence: 99%

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Agarwal

Klein

Knauer

et al. 2007

Discrete Comput Geom

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The detour and spanning ratio of a graph G embedded in E d measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n log n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we 18 Discrete Comput Geom (2008) 39: 17-37 obtain O(n log 2 n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in E 3 , and show that computing the detour in E 3 is at least as hard as Hopcroft's problem.

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Generalized self-approaching curves

Cited by 26 publications

References 7 publications

Comparison of Distance Measures for Planar Curves

Comparison of Distance Measures for Planar Curves

Self-approaching paths in simple polygons

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

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