On Envelopes of Arrangements of Lines

Eu, David; Guévremont, Eric; Toussaint, Godfried T.

doi:10.1006/jagm.1996.0040

Cited by 11 publications

(4 citation statements)

References 19 publications

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“…The natural formulation is that, if every sufficiently large subset of a family of contractible hulls has nonempty intersection, then the whole family has nonempty intersection. However, despite some formal similarities between similarly defined shapes and convex polygons [12], there can be no such result, as we now show.…”

Section: Analogues Of Helly's Theoremcontrasting

confidence: 59%

Regression Depth and Center Points

Amenta¹,

Bern²,

Eppstein³

et al. 2000

Discrete Comput Geom

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We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ⌈n/(d + 1)⌉, as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ⌈n/(d + 1)⌉ hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.

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Section: Analogues Of Helly's Theoremcontrasting

confidence: 59%

Regression Depth and Center Points

Amenta¹,

Bern²,

Eppstein³

et al. 2000

Discrete Comput Geom

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“…Recall that Bose et al [7] were the first to introduce line arrangement graphs and its definition resembles the one given by Eu, Grévremont and Toussaint [20] who gave an efficient algorithm for finding the envelope of a line arrangement. This problem was earlier studied by Ching and Lee [9] in 1985.…”

Section: Eccentricities In Pseudoline Arrangement Graphsmentioning

confidence: 99%

Pseudoline arrangement graphs: degree sequences and eccentricities

Das,

Rao,

Sahoo

2021

Preprint

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A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give a simple criterion based on the degree sequence that says whether a degree sequence will have a pseudoline arrangement graph as one of its realizations. In the second part, we study the eccentricities of vertices in such graphs. We observe that the diameter (maximum eccentricity of a vertex in the graph) of any pseudoline arrangement graph on n pseudolines is n − 2. Then we characterize the diametrical vertices (whose eccentricity is equal to the graph diameter) of pseudoline arrangement graphs. These results hold for line arrangement graphs as well.

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“…Although curiosity was our primary motivation, researchers from pattern recognition looking for concise descriptors of lines have studied hulls of random line arrangements [6,9], and "envelopes of lines," which are the union of all finite cells in line arrangements [8,10,12].…”

Section: Introductionmentioning

confidence: 99%