This paper studies the smoothness and the curvature of con ict sets of the dis tance function in the plane Con ict sets are also well known as bisectors We prove smoothness in the case of two convex sets and give a formula for the curvature We generalize moreover to weighted distance functions the so called Johnson Mehl model Introduction We consider two regions in the Euclidean plane A classic...

Starting with any nondegenerate triangle, we can use an interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) to any given tria...

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) any gi...

In this paper, we propose to compute Voronoi diagrams over mesh surfaces driven by an arbitrary geodesic distance solver, assuming that the input is a triangle as well collection of sites $P=\{p_i\}_{i=1}^m$ on surface. We two key techniques solve problem. First, partition determined minimizing $m$ fields, each which rooted at source site, suggest keeping one or more triples, for triangle, may ...

The single-lined spectroscopic binary $\nu$ Octantis provided evidence of the first conjectured circumstellar planet demanding an orbit retrograde to stellar orbits. planet-like behaviour is now based on 1437 radial velocities (RVs) acquired from 2001 2013. Oct's semimajor axis only 2.6 AU with candidate orbiting Oct A about midway between. These details seriously challenge our understanding fo...

The famous Erdös-Mordell inequality in geometric inequalities states the following: Let P be an interior point of a triangle ABC. Let R1, R2, R3 be the distances from P to the vertices A,B,C, and let r1, r2, r3 be the distance from P to the sides BC,CA,AB, respectively. Then R1 +R2 +R3 ≥ 2(r1 + r2 + r3), (1.1) with equality if and only if △ABC is equilateral and P is its center. This equality w...

We present HATNet observations of XO-5b, confirming its planetary nature based on evidence beyond that described in the announcement of Burke et al. (2008), namely, the lack of significant correlation between spectral bisector variations and orbital phase. In addition, using extensive spectroscopic measurements spanning multiple seasons, we investigate the relatively large scatter in the spectr...

The study was performed on neurons with direction selective (DS) receptive fields (RFs) in the primary visual cortex of the cat. Preferred directions (PDs) of these cells to a single light spot and a system of two identical light spots moving across the RF with a given angle between them were compared. Directional interactions appeared when the angles between the directions of the two moving sp...

1. INTRODUCTION. Among the theorems of plane geometry, a privileged position is held by those that are true in neutral geometry, that is, without either assuming or denying the parallel postulate. Such, for example, are the first twenty-eight propositions of Euclid's Elements, including the triangle congruences SAS (I.4), 1 SSS (I.8), ASA (I.26), and the base angles of an isosceles triangle are...