Clara I. Grima

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.

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.

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.

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 natures solutions to achieve epithelial bending. Our findings pave the way to understand the three-dimensional organization of epithelial organs.Cell arrangement in the plane of epithelia is well studied, but its three-dimensional packing is largely unknown. Here the authors model curved epithelia and predict that cells adopt a geometrical shape they call “scutoid”, resulting in different apical and basal neighbours, and confirm the presence of scutoids in curved tissues.

Stabbers of line segments in the plane

The problem of computing a representation of the stabbing lines of a set S of segments in the plane was solved by Edelsbrunner et al. We provide efficient algorithms for the following problems: computing the stabbing wedges for S, finding a stabbing wedge for a set of parallel segments with equal length, and computing other stabbers for S such as a double-wedge and a zigzag. The time and space complexities of the algorithms depend on the number of combinatorially different extreme lines, critical lines, and the number of different slopes that appear in S.

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.

Monochromatic geometric k-factors in red-blue sets with white and Steiner points

Abstract We study the existence of monochromatic planar geometric k-factors on sets of red and blue points. When it is not possible to find a k-factor we make use of auxiliary points: white points, whose position is given as a datum and which color is free; and Steiner points whose position and color is free. We present bounds on the number of white and/or Steiner points necessary and/or sufficient to draw a monochromatic planar geometric k-factor.

Motion Planning and Visibility Problems using the Polar Diagram

Motion planning and visibility problems are some of the most important topics studied in Computer Graphics, Computational Geometry and Robotics. There exits several and important results to these problems. We propose a new approach in this paper using a preprocessing in the plane, the polar diagram. The polar diagram can be considered as a plane tessellation with similar characteristics to the Voronoi Diagram. The Euclidean distance criterion is changed by the minimal angle criterion in this new approach. The advantage of using polar diagrams is an optimal computing preprocessing time and their immediate applications to angle problems as visibility or motion planning problems.

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

The original version of this Article contained an error in ref. 39, which incorrectly cited ‘Fristrom, D. & Fristrom, J. W. in The Development of Drosophila melanogaster (eds. Bate, M. & Martinez-Arias, A.) II, (Cold spring harbor laboratory press, 1993)’. The correct reference is ‘Condic, M.L, Fristrom, D. & Fristrom, J.W. Apical cell shape changes during Drosophila imaginal leg disc elongation: a novel morphogenetic mechanism. Development111: 23-33 (1991)’. Furthermore, the last sentence of the fourth paragraph of the introduction incorrectly omitted citation of work by Rupprecht et al. The correct citation is given below. These errors have now been corrected in both the PDF and HTML versions of the Article. Rupprecht, J.F., Ong, K.H., Yin, J., Huang, A., Dinh, H.H., Singh, A.P., Zhang, S., Yu, W. & Saunders, T.E. Geometric constraints alter cell arrangements within curved epithelial tissues. Mol. Biol. Cell 28, 3582-3594 (2017).

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. Draw monochromatic geometric k-factors of bicolored point sets using auxiliary points.We use two types of auxiliary points: Steiner points and white points.We provide algorithms for constructing those k-factors.We give bounds on the number of auxiliary points needed.