A new 2D tessellation for angle problems: The polar diagram

The new approach we propose in this paper is a plane partition with similar features to those of the Voronoi Diagram, but the Euclidean minimum distance criterion is replaced for the minimal angle criterion. The result is a new tessellation of the plane in regions called Polar Diagram, in which every site is owner of a polar region as the locus of points with smallest polar angle respect to this site. We prove that polar diagrams, used as preprocessing, can be applied to many problems in Computational Geometry in order to speed up their processing times. Some of these applications are the convex hull, visibility problems, and path planning problems

Dilation-free graphs in the l1 metric

The dilation-free graph of a planar point set S is a graph that spans S in such a way that the distance between two points in the graph is no longer than their planar distance. Metrically speaking, those graphs are equivalent to complete graphs; however they have far fewer edges when considering the Manhattan distance (we give here an upper bound on the number of saved edges). This article provides several theoretical, algorithmic, and complexity features of dilation-free graphs in the l1-metric, giving several construction algorithms and proving some of their properties. Moreover, special attention is paid to the planar case due to its applications in the design of printed circuit boards

Diagonal flips in outer-triangulations on closed surfaces

We show that any two outer-triangulations on the same closed surface can be transformed into each other by a sequence of diagonal flips, up to isotopy, if they have a sufficiently large and equal number of vertices

Transforming triangulations on non planar-surfaces

We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinde

Transforming Triangulations on Nonplanar Surfaces

We consider whether any two triangulations of a polygon or a point set on a nonplanar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder

K-Factores en nubes bicromáticas

Consideramos una colección de puntos bicromática y nos preguntamos cuántos puntos adicionales son necesarios considerar para asegurar la existencia de un k {factor. Dos tipos de puntos adicionales serán tratados: puntos de Steiner y puntos blancos (con posición prefijada pero no así su color

Scutoids are a geometrical solution to three-dimensional packing of epithelia

As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modelling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbours in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term “scutoid”. The detailed analysis of diverse tissues confirms that generation of apico-basal intercalations between cells is a common feature during morphogenesis. Using biophysical arguments, we propose that scutoids make possible the minimization of the tissue energy and stabilize three-dimensional packing. Hence, we conclude that scutoids are one of nature's solutions to achieve epithelial bending. Our findings pave the way to understand the three-dimensional organization of epithelial organs.España Ministerio de Ciencia y Tecnología BFU2013-48988-C2-1-P and BFU2016-8079

Compact Grid Representation of Graphs

A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families

Reporting Bichromatic Segment Intersections from Point Sets

In this paper, we introduce a natural variation of the problem of computing all bichromatic intersections between two sets of segments. Given two sets R and B of n points in the plane defining two sets of segments, say red and blue, we present an O(n2) time and space algorithm for solving the problem of reporting the set of segments of each color intersected by segments of the other color. We also prove that this problem is 3-Sum hard and provide some illustrative examples of several point configurations

Monochromatic geometric k-factors for bicolored point sets with auxiliary points

Given a bicolored point set S, it is not always possible to construct a monochromatic geometric planar k-factor of S. We consider the problem of finding such a k-factor of S by using auxiliary points. Two types are considered: white points whose position is fixed, and Steiner points which have no fixed position. Our approach provides algorithms for constructing those k-factors, and gives bounds on the number of auxiliary points needed to draw a monochromatic geometric planar k-factor of S