This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is O (not (n) log n). The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson-Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387...

We consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main result, we show that a logarithmic-time query for shortest paths between boundary points can be performed using Õ(n) preprocessing time and Õ(n) space where n i...

Given a convex polygonal robot B with k edges capable of translation and rotation, and a set of polygonal obstacles {A1, . . . , Am} each with ni sides and n = ∑ m i=1 ni, Leven and Sharir [1] prove that the number of critical placements of B at which it makes 3 simultaneous contacts with the obstacles but does not intersect their interior is O(knλ6(kn)), where λs(u) is the maximum length of a ...

In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to the multivariable case. In 2006, Green and Tao [13] considered Dickson’s conjecture in the multivariable case and gave direct...

Evidence exists for the presence of a specific facial phenotype in nonaffected first-degree relatives of persons with CL/P. An increased risk for olfactory dysfunction has also been reported in CL/P-relatives. These phenotypic features can probably be explained via the presence of CL/P-related susceptibility genes. We aimed at confirming the occurrence of these endophenotypic traits in first-de...

Given a finite collection S of geometric objects such as hyperplanes or spheres in R , the arrangement A(S) is the decomposition of R into connected open cells of dimensions 0, 1, . . . , d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances ...

W. Sierpinski publicized the following problem proposed by Werner Mnich in 1956: Are there three rational numbers whose sum and product are both one! In 1960, J. W. S. Cassels proved that there are no rationals that meet the Mnich condition. This paper extends the Mnich problem to fc-tuples of rationals whose sum and product are one by providing infinite solutions for all k > 3. It also provide...