Convex Position Research Articles

New results on production matrices for geometric graphs

We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. For instance, a formula for the number of connected geometric graphs with given root degree, drawn on a set of n points in convex position in the plane, is presented. Further, we find the characteristic polynomials and we provide a characterization of the eigenvectors of the production matrices.

On a conjecture of Karasev

Karasev conjectured that for any set of 3k lines in general position in the plane, which is partitioned into 3 color classes of equal size k, the set can be partitioned into k colorful 3-subsets such that all the triangles formed by the subsets have a point in common. Although the general conjecture is false, we show that Karasev's conjecture is true for lines in convex position. We also discuss possible generalizations of this result.

Colouring the Triangles Determined by a Point Set

Let P be a set of n points in general position in the plane. We study the chromatic number of the intersection graph of the open triangles determined by P. It is known that this chromatic number is at least n 3 27 + O(n 2 ), and if P is in convex position, the answer is n 3 24 + O(n 2 ). We prove that for arbitrary P, the chromatic number is at most n 3 19.259 + O(n 2 ).

Reconstruction of the path graph

Let P be a set of n≥5 points in convex position in the plane. The path graph G(P) of P is an abstract graph whose vertices are non-crossing spanning paths of P, such that two paths are adjacent if one can be obtained from the other by deleting an edge and adding another edge.We prove that the automorphism group of G(P) is isomorphic to Dn, the dihedral group of order 2n. The heart of the proof is an algorithm that first identifies the vertices of G(P) that correspond to boundary paths of P, where the identification is unique up to an automorphism of K(P) as a geometric graph, and then identifies (uniquely) all edges of each path represented by a vertex of G(P). The complexity of the algorithm is O(Nlog⁡N) where N is the number of vertices of G(P).

An Efficient Algorithm of the Planar 3-Center Problem for a set of the convex position points

The planar 3-center problem for a set S of points given in the plane asks for three congruent circular disks with the minimum radius, whose union can cover all points of S completely. In this paper, we present an O(n2 log3n) time algorithm for a restricted planar 3-center problem in which the given points are in the convex positions , i.e. The given points are the vertices of a convex polygon exactly.

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider maximizing the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a convex straight-line drawing, that is, a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that admits a non-convex drawing with more crossings than any convex drawing. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case. We also prove that the unweighted topological case is NP-hard.

Dependencies of the die-roll height during fine blanking of case hardening steel 16MnCr5 without V-ring using a nesting strategy

For economic reasons, fine blanking of asymmetric parts is performed with shifted and rotated part nesting. It was observed that both the part position and the part orientation on the sheet metal strip affect the die-roll height. Therefore, the influence of the bridge width between the produced cutting line and the strip edge or existing sheared edges was studied in this work. Additionally, the effects of the stroke number, the convexity of the radii and the sheet metal anisotropy were considered. The parts were manufactured in a production series of 40,000 parts with a fine blanking tool ejecting four parts simultaneously. From each nest, 20 parts were separated at four equidistant steps. The die-roll height was determined in three regions with different convexity radii of parts from varying nests. A normal distribution was found for the die-roll height, categorized by convexity radius and position of the nest. The die-roll height was found to be promoted by convexity radius, small bridge width, and in transverse blank direction. The number of strokes had a minor impact. Interaction effects occur particularly at small convexity radii and small bridge widths. As a conclusion, the nesting must be considered in order to reduce the die-roll height. Small bridge widths, small convexity radii, and the orientation in transverse blank direction have a negative impact, whereas the stroke number has a marginal influence. The understanding of the correlation between die-roll formation and nesting helps to reduce waste in mass manufacturing.

Competitive Online Routing on Delaunay Triangulations

Let [Formula: see text] be a graph, [Formula: see text] be a source node and [Formula: see text] be a target node. The sequence of adjacent nodes (graph walk) visited by a routing algorithm is a [Formula: see text]-competitive route if its length in [Formula: see text] is at most [Formula: see text] times the length of the shortest path from [Formula: see text] to [Formula: see text] in [Formula: see text]. We present a [Formula: see text]-competitive online routing algorithm on the Delaunay triangulation of an arbitrary given set of points in the plane. This improves the competitive ratio on Delaunay triangulations from the previous best of [Formula: see text]. We also present a [Formula: see text]-competitive online routing algorithm for Delaunay triangulations of point sets in convex position.

Estimating carbon stocks in young moraine soils affected by erosion

In this paper the storage potential of soils within a heterogeneous structured hummocky young moraine region for organic carbon is discussed with respect to climate change and erosion. Erosion is discussed to be either a global terrestrial CO2 sink or a source. In hummocky young moraine regions of North East Germany water and tillage erosion are steadily changing factors since the beginning of arable landuse in ancient times. For such topographically complex landscapes the knowledge on soil organic carbon (SOC) dynamics and the limits of carbon storage are still limited. Our objective is to combine data collected during former soil erosion studies with recent findings on (i) soil property and (ii) estimated “optimal” SOC data to predict the SOC storage related to tillage and crop rotation, among others. Classified catenae were analysed for texture, SOC, CO3-C, nutrient contents, and depth of weathering. Optimal SOC contents were estimated on the fine sized particle content. Arable soil at convex slope positions of steep catenae show 4 time smaller SOC stocks as compared to respective forest soils and to arable soils at concave position. Our findings suggest changes in SOC stocks to be almost exclusively related to decomposable carbon pools. Comparison of estimated optimal with measured SOC contents in soils at such positions indicated that such soils could potentially store a surplus of 0.6 to 0.8gkg−1. SOC protection at convex positions is limited by soil texture, and frequent truncation of the respective soil profiles. Whereas truncation followed by downhill transfer may bury SOC at sedimentation/concave positions resulting in long-term SOC storage as far as decomposition is prevented by site conditions.

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set S of n points in convex position in the plane into a data structure supporting queries of the following form: given a point q and a directed line $$\ell $$ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q among all points to the left of line $$\ell $$ . We present two data structures for this problem. The first data structure uses $$O(n^{1+\varepsilon })$$ space and preprocessing time, and answers queries in $$O(2^{1/\varepsilon }\log n)$$ time, for any $$0< \varepsilon < 1$$ . The second data structure uses $$O(n \log ^3 n)$$ space and polynomial preprocessing time, and answers queries in $$O(\log n)$$ time. These are the first solutions to the problem with $$O(\log n)$$ query time and $$o(n^2)$$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $$O(\log n)$$ amortized pointer changes, in addition to $$O(\log n)$$ -time point-location queries, even though every such update may make $$\Theta (n)$$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) amortized pointer changes per operation while keeping $$O(\log n)$$ -time point-location queries.

Incremental Voronoi Diagrams

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in \(\mathbb {R}^3\). We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is \(O(\sqrt{n})\). A matching \(\Omega (\sqrt{n})\) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the \(O(\log {n})\) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of \(S \cup \{p\}\) from the diagram of S, in time \(O(K\,\mathrm {polylog}\ n)\) worst case, which is \(O(\sqrt{n}\;\mathrm {polylog}\ n)\) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in \(O(\log n)\) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

Characteristic polynomials of production matrices for geometric graphs

An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.

Diagnostic value of profile photography in identifying departures from the norm of growth pattern and horizontal jaw relationship.

Although the importance of facial photography in orthodontics has been recognized, there is only one study on its validity and absolutely no studies on its diagnostic value. This study was conducted, therefore, to evaluate for the first time the diagnostic merit of photography in detecting growth pattern and jaw relationships. Standardized lateral cephalographs and profile photographs were taken from 54girls and 26boys. On cephalometry, growth pattern (normal, horizontal, and vertical) and horizontal bimaxillary classification (Classes I to III) were determined. Using photographs, the angles FH-MP', CP'-MP', Sn-T-Me', cutaneous analog of ANB, facial convexity, and head position were calculated. Diagnostic photography-based measurements for detecting departures from the norm were calculated by counting correct and incorrect diagnoses, according to the gold standard (lateral cephalometry). The sensitivity, specificity and positive predictive values, and the negative predictive value of FH-MP' photographic measurements in detecting growth disorders were 51.4, 77.8, 64.3, and 67.3 percent, respectively. These values were 45.7, 95.6, 88.9, and 69.4 percent for the CP'-MP' measurement, and were 40.0, 93.3, 82.4, and 66.7 percent for Sn-T-Me', and 51.4, 100, 100, and 72.6 percent for all of them combined. In detecting the horizontal relationship between the jaws, the photographic method showed an overall sensitivity, specificity, PPV, and NPV of 80.9, 63.6, 76.0, and 70.0 percent, respectively. These values were 80.9 57.6 73.1 67.9 percent for cutaneous analog of ANB, 87.2, 45.5, 69.5, and 71.4 percent for facial convexity, and 61.7, 66.7, 72.5, and 55.0 percent for head position. Profile photography can be used as a perfectly accurate measurement tool for detecting horizontal and vertical growth patterns. It is also a rather accurate method for detecting the normal growth pattern, as well as both Class I and non-Class I jaw relationships. These results highlight this non-invasive and economic method as an effective clinical diagnostic tool.

A SAT attack on the Erdős–Szekeres conjecture

A classical conjecture of Erdős and Szekeres states that, for every integer k≥2, every set of 2k−2+1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2k−2+1 vertices contains an ordered subhypergraph with k vertices and k−2 edges, which is a union of a red monotone path and a blue monotone path that are vertex disjoint except for their two common end-vertices.Applying a state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.

On Hamiltonian alternating cycles and paths

We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n2) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.

Faster bottleneck non-crossing matchings of points in convex position

Given an even number of points in a plane, we are interested in matching all the points by straight line segments so that the segments do not cross. Bottleneck matching is a matching that minimizes the length of the longest segment. For points in convex position, we present a quadratic-time algorithm for finding a bottleneck non-crossing matching, improving upon the best previously known algorithm of cubic time complexity.

Effects of convex cavity structure, position and number on conversion of methane catalytic combustion and extinction limit in a micro-channel: A numerical study

The micro-channel combustion increasingly plays a significant role in Micro-Electro-Mechanical System (MEMS). The numerical investigation has been carried out in a micro-channel with detailed reaction mechanism. Also, an effective method has been identified to improve methane conversion efficiency. Moreover, the extinction limit of premixed methane/air catalytic combustion has been determined with the presence of a convex cavity. And, the effects of four factors for normalized cavity length, normalized cavity diameter, position, and number of convex cavities on the catalytic combustion characteristics of methane/air in a micro-channel have been investigated with numerical method, respectively. Comparison of the above four factors, increasing the normalized cavity diameter of the convex cavity is the most effective way to improve methane conversion ratio and expand extinction limit. The results also show that a vortex street in the channel appears with normalized cavity diameter of 0.3, the recirculation zone and vortex street zone in the channel improve the stability of combustion. Besides, the methane conversion ratio can be increased to 90.3% and the extinction limit can be reached up to 18.9m/s at normalized cavity diameter of 0.3. This study provides some useful information for the design and operation of the micro-combustor with convex cavity.

Towards plane spanners of degree 3

Let $S$ be a finite set of points in the plane. In this paper we consider the problem of computing plane spanners of degree at most three for $S$. If $S$ is in convex position, then we present an algorithm that constructs a plane $\frac{3+4\pi}{3}$-spanner for $S$ whose vertex degree is at most 3. If $S$ is the vertex set of a non-uniform rectangular lattice, then we present an algorithm that constructs a plane $3\sqrt{2}$-spanner for $S$ whose vertex degree is at most 3. If $S$ is in general position, then we show how to compute plane degree-3 spanners for $S$ with a linear number of Steiner points.

Degree four plane spanners: Simpler and better

Let $\mathcal{P}$ be a set of $n$ points embedded in the plane, and let $\mathcal{C}$ be the complete Euclidean graph whose point-set is $\mathcal{P}$. Each edge in $\mathcal{C}$ between two points $p$, $q$ is realized as the line segment $[pq]$ and is assigned a weight equal to the Euclidean distance $|pq|$. In this paper, we show how to construct in $O(n \lg n)$ time a plane spanner of $\mathcal{C}$ of maximum degree at most $4$ and of stretch factor at most $20$. This improves a long sequence of results on the construction of bounded degree plane spanners of $\mathcal{C}$. Our result matches the smallest known upper bound of $4$ by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from $156.82$ to $20$. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner and reveals useful structural properties of Delaunay triangulations defined with respect to the equilateral-triangle distance. The structure of the constructed spanner implies that when $\mathcal{P}$ is in convex position, the maximum degree of the spanner is at most $3$. Combining the above degree upper bound with the fact that $3$ is a lower bound on the maximum degree of any plane spanner of $\mathcal{C}$ when the point-set $\mathcal{P}$ is in convex position, the results in this paper give a tight bound of $3$ on the maximum degree of plane spanners of $\mathcal{C}$ for point-sets in convex position.

The probability that $n$ random points in a disk are in convex position

Pick $n$ random points $x_{1},\dots,x_{n}$ uniformly and independently in a disk and consider their convex hull $C$. Let $P_{D}^{n,m}$ be the probability that exactly $m$ points among the $x_{i}$’s are on the boundary of the convex hull of $\{x_{1},\ldots,x_{n}\}$ (so that $P_{D}^{n,n}$ is the probability that the $x_{i}$’s are in a convex position). In the paper, we provide a formula for $P_{D}^{n,m}$.