An algorithmic approach to convex fair partitions of convex polygons

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. The existence of such a partition for any number of regions remains an open question. In this paper, we address this issue by developing an algorithm to find such a convex fair partition without restrictions on the number of regions. Our approach utilizes the normal flow algorithm (a generalization of Newton’s method) to find a zero for the excess areas and perimeters of the convex hulls of the regions. The initial partition is generated by applying Lloyd’s algorithm to a randomly selected set of points within the polygon, after appropriate scaling. We performed extensive experimentation, and our algorithm can find a convex fair partition for 100% of the tested problem. Our findings support the conjecture that a convex fair partition always exists.

Classical dissections convert any planar polygonal region onto any other polygonal region having the same area. If two convex polygonal regions are isoparametric, that is, have equal areas and equal perimeters, our main result states that there is always a dissection, called a complete dissection, that converts not only the regions but also their boundaries onto one another. The proof is constructive and provides a general method for complete dissection using frames of constant width. This leads to a new object of study: isoparametric polygonal frames, for which we show that a complete dissection of one convex polygonal frame onto any other always exists. We also show that every complete dissection can be done without flipping any of the pieces.

A cyclic quadrilateral is called a Brahmagupta quadrilateral if the lengths of its four sides and two diagonals, and the area are all given by integers. In this paper, we consider the hitherto unsolved problem of finding two Brahmagupta quadrilaterals with equal perimeters and equal areas. We obtain two parametric solutions of the problem — the first solution generates examples in which each quadrilateral has two equal sides while the second solution gives quadrilaterals all of whose sides are unequal. We also show how more parametric solutions of the problem may be obtained.

PurposeThe purpose of this paper is to investigate the performance of Passive Direct Methanol Fuel Cell (PDMFC) experimentally using various Membrane Electrode Assembly (MEA) shapes such as square, rectangle, rhombus, and circle with equal areas and equal perimeters. The variation in MEA shape/size is achieved by altering gasket openings in the dynamic regions.Design/methodology/approachIn the equal areas of MEA shapes, gasket opening areas of 1963.5 (+/−0.2) mm2 are used. Whereas in the equal perimeters of shapes, gasket opening perimeters of 157.1 (+/−0.2) mm are used. In this experimentation, Nickel-201 current collectors with 45.3% of circular openings are used on both the anode and cathode sides. The experiment is carried out at a 5 molar methanol concentration to find out the highest power density of the cell.FindingsIn the equal areas, among the shapes that are chosen for investigation, the square shape opening consisting of a perimeter of 177.2 mm has developed a maximum power density of 6.344 mWcm−2 and a maximum current density of 65.2 mAcm−2. Similarly, in equal perimeters, the rhombus shape opening with an area of 1400 mm2 has developed a maximum power density of 7.714 mWcm−2 and a maximum current density of 85.3 mAcm−2.Originality/valueThe novelty of this research work is instead of fabricating various shapes and sizes of highly expensive MEAs, the desired shapes and sizes of the MEA are achieved by altering gasket openings over dynamic regions to find out the highest power density of the cell.

Much has been written on triangles having areas and perimeters of equal numerical value. Markowitz (1981) determined that only five triangles having sides with integral lengths exist for which the areas equal the perimeters, but that infinitely many have rational side lengths. (Throughout this article, when it is stated that triangles have equal area and perimeter, it means that they are numerically equal.) Markowitz (1981) and Bates (1979) both determine equations satisfied by triangles having equal areas and perimeters, but these equations can only be used through trial and error. The purpose of this article is to find a method that will easily generate any number of triangles of equal area and perimeter, and also geometrically characterize such triangles.

In this paper we study the problem of partitioning a convex polygon \(P\) with \(n\) vertices into \(m\) polygons of equal area and perimeter and give efficient algorithms for \(m = 2\) and for the more general case when \(m\) is a power of 2. While it was known such a partition exists, no algorithmic results were published so far. Our algorithm for \(m = 2\) is optimal and runs in \(O(n)\) time, while the algorithm for \(m = 2^k\), where \(k \ge 2\) is an integer, runs in \(O(n (2n)^{k-1})\) time. The algorithms have been implemented and tested on randomly generated convex polygons.KeywordsConvex PolygonFair PartitionEqual AreaArea BisectorsCake-cutting ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Algorithms for fair partitioning of convex polygons

In this study, the researchers address the conundrum: To divide a 2-dimensional convex polygonal region Q into a specified natural number n of convex pieces, all of which have equal areas although perimeters could be different, such that the total length of cuts used is maximized. With the aid of AutoCAD software, the results show that a convex polygon Q divided into n=2 equal area partitions must have its cut line λ¬ or its area bisector originate in some vertex "v" _"α" of Q with the least measure. We claim that this cut line λ is the maximum possible cut length that divides polygon Q into n=2 equal area partitions. The resulting proof of this conjecture can be used recursively for a convex polygon Q divided into n=2^k equal area partitions, where k∈N

LetT be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node inT is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the con- structed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may dier slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper- rectangular partitions in R d . We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a polylog()-approx

Tower foundations have been used as grounding electrodes to reduce the area of the grounding devices. However, it is difficult to calculate the grounding resistance due to the complex structure of the reinforced concrete foundation. A method for calculating grounding resistance of reinforced concrete foundations is proposed in this paper. The method equates the complex foundation structure into a cylindrical conductor and then calculates the grounding resistance with the help of the method of moments, which simplifies establishment of the simulation model and reduces the extensive computation. In addition, the applicability of the equal cross-sectional area method and the equal cross-sectional perimeter method is analyzed. It shows that both methods are applicable only when the concrete resistivity is close to the soil resistivity. The equal cross-sectional area method is applicable when the concrete resistivity is within twice the soil resistivity, while the equal cross-sectional perimeter method is applicable when the concrete resistivity is approximately same or less than the resistivity of the soil.

Equipartitioning by a convex 3-fan

A promising approach of caved vacuum insulation panel and investigation on its thermal bridge effect

NHỮNG HÌNH CÓ CÙNG DIỆN TÍCH, CHU VI TRONG SÁCH GIÁO KHOA TOÁN TIỂU HỌC VIỆT NAM VÀ PHÁP

The effect of nanopores geometry on desalination of single-layer graphene-based membranes: A molecular dynamics study

We give a fast and simple factor 2.74 approximation algorithm for the problem of choosing the k medians of the continuum of demand points defined by a convex polygon C. Our algorithm first surrounds the input region with a bounding box, then subdivides the bounding box into subregions with equal area. Simulation results on the convex hulls of the 50 states in the United States show that the practical performance of our algorithm is within 10% of the optimal solution in the vast majority of cases.