Star-shaped Polygon Research Articles

Counting Carambolas

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.

Technical and functional analysis of Albolafia waterwheel (Cordoba, Spain): 3D modeling, computational-fluid dynamics simulation and finite-element analysis

A detailed study has been made of a vertical waterwheel, the wheel of Albolafia situated on the Guadalquivir river near the city of Cordoba (Spain). We propose a methodology for ad hoc research based on three aspects: 3D geometric modeling, analysis with computational fluid-dynamics techniques and dynamic simulation of the whole and its finite-element analysis. The results show the correct operation of the waterwheel with an initial moment of inertia of 90,800Nm and a range of water-flow speeds of between 0.91 and 1.01m/s. These values are related to the average flow of the river, which allowed the wheel to operate at least 124days per year. The spatial distribution of stresses has shown that the full buckets created an imbalance compared with the empty ones, and that the star-shaped polygon reinforcement effectively absorbed these tensions. In addition, the oak wood used in the construction of the waterwheel proved highly resistant, as the maximum working stress has never been surpassed, reflecting the effectiveness of the materials used at the time.

Compaction kernel geomerty ensamble projection on medium atomic structure

This article describes the results of analysis of compaction kernel occurrence when medium destruction. Also it considers problems of identification of compaction kernel geometry (height) dependence upon state and physical and mathematical characteristics of medium atomic structure

Minimum cost star-shaped drawings of plane graphs with a fixed embedding and concave corner constraints

A star-shaped drawing of a plane graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, given a biconnected plane graph G with fixed plane embedding and a prescribed set of concave corners, we study the following two problems on star-shaped drawings.First, we consider the problem of finding a star-shaped drawing D of G such that only prescribed corners are allowed to become concave corners in D. More specifically, we characterize a necessary and sufficient condition for a subset of prescribed corners to admit such a star-shaped drawing D of G. Our characterization includes Thomassen’s characterization of biconnected plane graphs with a prescribed boundary that have convex drawings [24]. We also give a linear-time testing algorithm to test such conditions.Next, given a non-negative cost for each corner in G, we show that a star-shaped drawing D of G with the minimum cost can be found in linear-time, where the cost of a drawing is defined by the sum of costs of concave corners in the drawing.

A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners

A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, we consider the problem of finding a star-shaped drawing of a biconnected planar graph with the minimum number of concave corners. We first show new structural properties of planar graphs to derive a lower bound on the number of concave corners. Based on the lower bound, we prove that the problem can be solved in linear time by presenting a linear-time algorithm for finding a best plane embedding of a biconnected planar graph with the minimum number of concave corners. This is in spite of the fact that a biconnected planar graph may have an exponential number of different plane embeddings.

A combinatorial property on angular orders of plane point sets

We study the following combinatorial property of point sets in the plane: For a set S of n points in general position and a point p ∈ S consider the points of S − p in their angular order around p. This gives a star-shaped polygon (or a polygonal path) with p in its kernel. Define c ( p ) as the number of convex angles in this star-shaped polygon around p, and c ( S ) as the sum of all c ( p ) , for p ∈ S . We show that for every point set S, c ( S ) is always at least 1 2 n 3 2 − O ( n ) . This bound is shown to be almost tight. Consequently, every set of n points admits a star-shaped polygonization with at least n 2 − O ( 1 ) convex angles.

COMPUTING SIGNED PERMUTATIONS OF POLYGONS

Given a planar polygon (or chain) with a list of edges {e1, e2, e3, …, en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.

The effects of hydrogen substitution on turbulent premixed methane–air kernels using direct numerical simulation

Abstract Direct numerical simulations (DNS) are used to assess the effects of hydrogen substitution on lean premixed methane–air kernels during the early stages of growth in freely decaying turbulence. Two-dimensional simulations with a detailed 68-step reaction mechanism are carried out at equivalence ratios of ϕ = 0.53 and ϕ = 0.625, both with and without the substitution of a 30% fuel mole fraction of hydrogen. A comparative analysis is made into the changes in turbulent flame speeds, global stretch rates, and flame wrinkling at different turbulence intensities. The underlying causes of these changes are investigated through the distributions of the surface-conditioned displacement speed, strain rate and curvature. Direct comparison is made with the planar flame results of Hawkes and Chen [1] to assess the qualitative effects of kernel geometry in combination with a hydrogen-enriched fuel. It was found that the reduced effective fuel Lewis number and preferential diffusion of hydrogen, combined with the higher stretch rates and mean positive curvature of the kernel make the effects of hydrogen enrichment much more pronounced in kernels compared to planar flames.

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing. In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.

단순 다각형 계층구조에서의 삼각화와 경비가능충분집합

본 논문은 화랑문제 분야에 관한 것으로, 다각형의 계층구조에서 경비충분집합에 될 수 있는 기하학적인 요소들에 관해 다루었다. 경비충분 집합이 될 수 있는 기하학적인 요소로 다각형의 삼각화를 고려하였고, 다각형의 삼각화한 대각선분에 대해 완전가시성으로 양쪽을 다 감시할 경우 경비충분집합이 되는 삼각형의 부류가 볼록 다각형, 단변단조 다각형, 소용돌이 다각형임을 보였고, 그 외의 별모양 다각형, 단조 다각형, 완전외부가시성 다각형에서는 경비충분집합이 되지 못함을 보였다. In this paper, we consider a characterization of a Guard Sufficiency Set(GSS) in the hierarchy of simple polygons. we propose the diagonals of a arbitrary triangulation of a polygon as a GSS when guards see the diagonals with completely visibility and both sides of the diagonal. we show that this can be a GSS in convex polygons, unimodal polygons, spiral polygons but this can not be a GSS in star-shaped polygons, monotone polygons, completely external visible polygons.

Parallel MRI with extended and averaged GRAPPA kernels (PEAK‐GRAPPA): Optimized spatiotemporal dynamic imaging

To evaluate an optimized k-t-space related reconstruction method for dynamic magnetic resonance imaging (MRI), a method called PEAK-GRAPPA (Parallel MRI with Extended and Averaged GRAPPA Kernels) is presented which is based on an extended spatiotemporal GRAPPA kernel in combination with temporal averaging of coil weights. The PEAK-GRAPPA kernel consists of a uniform geometry with several spatial and temporal source points from acquired k-space lines and several target points from missing k-space lines. In order to improve the quality of coil weight estimation sets of coil weights are averaged over the temporal dimension. The kernel geometry leads to strongly decreased reconstruction times compared to the recently introduced k-t-GRAPPA using different kernel geometries with only one target point per kernel to fit. Improved results were obtained in terms of the root mean square error and the signal-to-noise ratio as demonstrated by in vivo cardiac imaging. Using a uniform kernel geometry for weight estimation with the properties of uncorrelated noise of different acquired timeframes, optimized results were achieved in terms of error level, signal-to-noise ratio, and reconstruction time.

Mathematical modelling of wheat kernel drying with input from moisture movement studies using magnetic resonance imaging (MRI), Part II: Model comparison with published studies

The predicted moisture data, using the three-dimensional (3D) single wheat drying model (Part I) considering actual kernel geometry, were compared with two published semi-empirical thin-layer wheat drying models, and one published axisymmetric heat and mass transfer model for a single wheat kernel. The published axisymmetric model assumed a uniform initial moisture distribution within the whole kernel of ellipsoidal geometry and a single value of the water diffusion coefficient. Our model predictions agreed well with the observations obtained from the published models. Further, our model can predict the moisture distribution pattern within individual wheat components with drying time. Moisture and temperature distributions inside the wheat kernel and the effect of drying temperature on wheat quality were explained by calculating the Biot and Lewis numbers. High Biot numbers for moisture transfer (0.89×106–3.04×106 for endosperm and 1.98×106–3.37×106 for germ) indicated that moisture transfer inside the wheat structural components was governed by high internal resistance, whereas low Biot numbers for heat transfer (0.12–0.20 for endosperm and 0.02–0.10 for germ) indicated that heat transfer was controlled by external resistance and, hence, temperature within the wheat kernels was uniform at any given time during drying. Lewis numbers indicated that the rate of heat transfer was very high compared to moisture transfer in both the endosperm and the germ. Sensitivity analysis was performed for selecting proper model parameters which indicated that values of water diffusion coefficients in the germ and endosperm are the most sensitive parameters for a grain drying process.

Convex drawings of graphs with non-convex boundary constraints

In this paper, we study a new problem of convex drawing of planar graphs with non-convex boundary constraints, and call a drawing in which every inner-facial cycle is drawn as a convex polygon an inner-convex drawing. It is proved that every triconnected plane graph with the boundary fixed with a star-shaped polygon whose kernel has a positive area admits an inner-convex drawing. We also prove that every four-connected plane graph whose boundary is fixed with a crown-shaped polygon admits an inner-convex drawing. We present linear time algorithms to construct inner-convex drawings for both cases.

Note on covering monotone orthogonal polygons with star-shaped polygons

In 1986, Keil provided an O ( n 2 ) time algorithm for the problem of covering monotone orthogonal polygons with the minimum number of r-star-shaped orthogonal polygons. This was later improved to O ( n ) time and space by Gewali et al. in [L. Gewali, M. Keil, S.C. Ntafos, On covering orthogonal polygons with star-shaped polygons, Information Sciences 65 (1992) 45–63]. In this paper we simplify the latter algorithm—we show that with a little modification, the first step Sweep1 of the discussed algorithm—which computes the top ceilings of horizontal grid segments—can be omitted. In addition, for the minimum orthogonal guard problem in the considered class of polygons, our approach provides a linear time algorithm which uses O ( k ) additional space, where k is the size of the optimal solution—the algorithm in [L. Gewali, M. Keil, S.C. Ntafos, On covering orthogonal polygons with star-shaped polygons, Information Sciences 65 (1992) 45–63] uses both O ( n ) time and O ( n ) additional space.

Watched guards in art galleries

The art gallery problem asks how many guards are sufficient to see every point of the interior of a polygon. A set of guards is called watched if each guard itself is seen by at least one of its colleagues. In 1994, Hernandez-Penalver wrote that \( \lfloor {\frac{{2n}} {5}} \rfloor \)watched guards always suffice to guard any polygon with n vertices. However in 2001, Michael and Pinciu, and independently Żylinski, presented a class of polygons that required more than \( \lfloor {\frac{{2n}} {5}} \rfloor \)watched guards – which disproved the Hernandez-Penalver’s result – and they established a new tight bound for watched guards: \( \lfloor {\frac{{3n - 1}} {7}} \rfloor \). Combinatorial bounds for watched guards in orthogonal polygons were independently given by Hernandez-Penalver , and by Michael and Pinciu, who proved the \( \lfloor {\frac{{n}} {3}} \rfloor \) -bound to be tight. In this paper, tight bounds for polygons of miscellaneous shapes are presented: \( \lfloor {\frac{{2n}} {5}} \rfloor \) watched guards for monotone and spiral polygons, and \( \lfloor {\frac{{3n - 1}} {7}} \rfloor \) vertex watched guards for star-shaped polygons.

CONSTRUCTION OF A 3-D FINITE ELEMENT MODEL OF THE HUMAN FOREARM COMPLEX USING A SERIES OF 2-D COLOR IMAGES

A simple yet efficient paradigm for geometric mesh generation using the Visible Human Project Male dataset for further finite element analysis was presented. The minimum distance classifier was used for the discriminant function between the class centers classified by the fuzzy c-means clustering method in the RGB space. Furthermore, based on two major geometric assumptions on the boundary curves, star-shaped polygon and geometric conformity, a points-on-line search technique was devised for efficient computation of the boundary points of the contours for each anatomical component of the human forearm complex. The computed boundary points in each slice were fitted to a closed spline curve and resampled and then refitted for correct alignment with the consecutive boundary curves in order to improve geometric fidelity. By using the refitted contours, a 3-D geometric model of the human radius, ulna, and surrounding soft tissue was generated in a commercial computer–aided design system and exported to a commercial finite element analysis package for meshing with its built-in automatic mesh generator. The proposed method can be applied to geometric mesh generation of other long bones, which allows easy handling, storage, and exchange of the model.

Modeling of Diffusion in Ellipsoidal Solids: A Comparative Study

Most food materials are usually measured in three mutually perpendicular dimensions (length, width, and thickness) and considered approximately as general ellipsoids. In this article, we briefly reviewed the available studies on how to deal with the diffusion problems in ellipsoidal solids. A three-dimensional (3-D) model was proposed to accurately represent the kernel geometry and rigorously interpret the diffusion phenomena. By the finite difference method, the numerical solution of the diffusion equation for kernels of different shape factors was obtained and compared with those predicted by the one-dimensional (sphere) model and two-dimensional (axi-symmetrical body) model. Meanwhile, the effectiveness of the latter two models was evaluated.

Searching and on-line recognition of star-shaped polygons

We study the problem of on-line searching for a target inside a polygon. In particular, we propose a strategy for finding a target of unknown location in a star-shaped polygon with a competitive ratio of 11.52. We also provide a lower bound of 9 for the competitive ratio of searching in a star-shaped polygon which is close to the upper bound. A similar task is the on-line recognition of a star-shaped polygon P . Here, the robot travels on a path that allows it to decide whether P is star-shaped or not. We present a strategy with a competitive ratio of 28.85 and give a lower bound of 82 for this problem.

PLANAR STRONG VISIBILITY

Strong visibility is a generalization of standard visibility, defined with respect to a fixed set of line orientations. We investigate computational properties of this generalized visibility, as well as the related notion of strong convexity, and describe algorithms for the following tasks: 1. Testing the strong visibility of two points in a polygon. 2. Finding the strong convex hull of a point set or polygon. 3. Constructing the strong kernel of a polygon. 4. Identifying the points that are strongly visible from a given point.

OPTIMAL POLYGON COVER PROBLEMS AND APPLICATIONS

Polygon cover problems are important in several applied areas, such as material layout, layered manufacturing, radiation therapy and radiosurgery, etc. In this paper, we study three optimal polygon cover problems: monotone polygon cover among obstacles, star-shaped polygon cover among obstacles, and strip cover for trapezoidalized polygons. Based on some interesting geometric observations, we develop efficient algorithms for solving these problems. The complexity bounds of our monotone cover and star-shaped cover algorithms are comparable to those of the previously best known algorithms for simpler cases of the problems without considering obstacles. Our strip cover algorithm improves the quality of the previously best known solutions.