This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from elementary geometry as dynamic entities: while the free points of a construction perform a continuous motion the dependent points should move consistently and continuously. We focus on constructions that are entirely built up...

We study the temporal evolution of downflows observed at the lateral edges of penumbral filaments in a sunspot located very close to the disk center. Our analysis is based on a sequence of nearly diffraction-limited scans of the Fe I 617.3 nm line taken with the CRisp Imaging Spectro-Polarimeter at the Swedish 1 m Solar Telescope. We compute Dopplergrams from the observed intensity profiles usi...

For Wireless Sensor Networks (WSNs), the Voronoi partition of a region is a challenging problem owing to the limited sensing ability of each sensor and the distributed organization of the network. In this paper, an algorithm is proposed for each sensor having a limited sensing range to compute its limited Voronoi cell autonomously, so that the limited Voronoi partition of the entire WSN is gene...

Our aim is to confirm the nature of the long period radial velocity measurements for β Gem first found by Hatzes & Cochran (1993). We present precise stellar radial velocity measurements for the K giant star β Gem spanning over 25 years. An examination of the Ca II K emission, spectral line shapes from high resolution data (R = 210,000), and Hipparcos photometry was also made to discern the tru...

Hydrogen atom positions of nucleotide bases in RNA structures solved by X-ray crystallography are commonly derived from heavy-atom coordinates by assuming idealized geometries. In particular, N1-H1 vectors in G and N3-H3 vectors in U are commonly positioned to coincide with the bisectors of their respective heavy-atom angles. We demonstrate that quantum-mechanical optimization of the hydrogen p...

We give a short trigonometric proof of the Steiner-Lehmus theorem. The well known Steiner-Lehmus theorem states that if the internal angle bisectors of two angles of a triangle are equal, then the triangle is isosceles. Unlike its trivial converse, this challenging statement has attracted a lot of attention since 1840, when Professor Lehmus of Berlin wrote to Sturm asking for a purely geometric...

Given a set of points in IR , called sites, we consider the problem of approximating the Voronoi cell of a site by a convex polyhedron with a small number of facets or, equivalently, of finding a small set of approximate Voronoi neighbors of . More precisely, we define an -approximate Voronoi neighborhood of , denoted , to be a subset of satisfying the following property: is an -approximate nea...

For each pair (Qi, Qj) of reference points and each real number r there is a unique hyperplane h ⊥ QiQj such that d(P,Qi)2 − d(P,Qj)2 = r for points P in h. Take n reference points in d-space and for each pair (Qi, Qj) a finite set of real numbers. The corresponding perpendiculars form an arrangement of hyperplanes. We explore the structure of the semilattice of intersections of the hyperplanes...

We present a fast and versatile algorithm for the reconstruction of the grain structure from 2d and 3d Electron Back Scatter Diffraction (EBSD) data. The algorithm is rigorously derived from the modeling assumption that grain boundaries are located at the bisectors of adjacent measurement locations. This modeling assumption immediately implies that grains are composed of Voronoi cells correspon...

We have observed the spectrum of Procyon A (F5IV) from 4559 to 5780 Å with a S/N of ∼ 10 and a resolving power of 2 × 10. We have measured the line bisectors and relative line shifts of a large number of Fe I and Fe II lines, comparing them to those found in the Sun. A three-dimensional (3D) hydrodynamical model atmosphere has been computed and is tested against observations. The model reproduc...