Proximity problems on line segments spanned by points

Proximity Problems on Line Segments Spanned by Points

Finding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(n logn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum)...

Finding Segments and Triangles Spanned by Points in R3

Comment on "Volumes spanned by random points in the hypercube"

Consider the hypercube [0, 1]n in R. This has 2n vertices and volume 1. Pick N = N(n) vertices independently at random, form their convex hull, and let Vn be its expected volume. How large should N(n) be to pick up significant volume ? Let κ = 2/ √ e ≈ 1.213, and let 2 > 0. We shall show that, as n →∞, Vn → 0 if N(n) ≤ (κ− 2)n, and Vn → 1 if N(n) ≥ (κ+ 2)n. A similar result holds for sampling u...

Grouping Edge Points into Line Segments by Sequential Hough Transformation

An algorithm to group edge points into digital line segments with Hough transformation is described in this paper. The edge points are mapped onto the parameter domain discretized at specific intervals, on which peaks appear to represent different line segments. By modeling each peak as a Gaussian function in the parameter domain, a region, to which the edge points are supposed to be mapped, is...

Selecting distances in arrangements of hyperplanes spanned by points

In this paper we consider a problem of distance selection in the arrangement of hyperplanes induced by n given points. Given a set of n points in d-dimensional space and a number k, 1 k (n d ) , determine the hyperplane that is spanned by d points and at distance ranked by k from the origin. For the planar case we present an O(n log2 n) runtime algorithm using parametric search partly different...