Topic
Distance from a point to a line
About: Distance from a point to a line is a research topic. Over the lifetime, 284 publications have been published within this topic receiving 5265 citations.
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TL;DR: It is shown that skeletons can be produced by simple procedures and since these are based on Euclidean distances it is assumed that they are superior to skeletons based on d4−, d8−, and even octagonal metrics.
1,777 citations
29 Mar 2015
TL;DR: In this paper, the authors proposed the use of modified distance calculation in generational distance and inverted generational distance (IGD) to evaluate the quality of an obtained solution set in comparison with a pre-specified reference point set.
Abstract: In this paper, we propose the use of modified distance calculation in generational distance (GD) and inverted generational distance (IGD). These performance indicators evaluate the quality of an obtained solution set in comparison with a pre-specified reference point set. Both indicators are based on the distance between a solution and a reference point. The Euclidean distance in an objective space is usually used for distance calculation. Our idea is to take into account the dominance relation between a solution and a reference point when we calculate their distance. If a solution is dominated by a reference point, the Euclidean distance is used for their distance calculation with no modification. However, if they are non-dominated with each other, we calculate the minimum distance from the reference point to the dominated region by the solution. This distance can be viewed as an amount of the inferiority of the solution (i.e., the insufficiency of its objective values) in comparison with the reference point. We demonstrate using simple examples that some Pareto non-compliant results of GD and IGD are resolved by the modified distance calculation. We also show that IGD with the modified distance calculation is weakly Pareto compliant whereas the original IGD is Pareto non-compliant.
255 citations
TL;DR: An optimal O( n log n ) algorithm for translating two sets of points on a line so that their Hausdorff distance is as small as possible.
233 citations
TL;DR: The Euclidean distance between syntactically linked words in sentences predicts, under ideal conditions, an exponential distribution of the distance between linked words, a trend that can be identified in real sentences.
Abstract: We study the Euclidean distance between syntactically linked words in sentences. The average distance is significantly small and is a very slowly growing function of sentence length. We consider two nonexcluding hypotheses: (a) the average distance is minimized and (b) the average distance is constrained. Support for (a) comes from the significantly small average distance real sentences achieve. The strength of the minimization hypothesis decreases with the length of the sentence. Support for (b) comes from the very slow growth of the average distance versus sentence length. Furthermore, (b) predicts, under ideal conditions, an exponential distribution of the distance between linked words, a trend that can be identified in real sentences.
131 citations