Minimum Perimeter Research Articles

The Orthic Triangle and the O.K. Quadrilateral

(2002). The Orthic Triangle and the O.K. Quadrilateral. The American Mathematical Monthly: Vol. 109, No. 8, pp. 704-728.

Triangles of Minimal Perimeter Circumscribed about a Convex n -Gon

An algorithm for finding a circumscribed triangle of the minimal perimeter is suggested. Properties of such a triangle are described. Bibliography: 1 title.

Minimum perimeter partitions of the plane into equal numbers of regions of two different areas

We identify the minimum-perimeter periodic tilings of the plane by equal numbers of regions (cells) of areas 1 and λ (minimal tilings), with at most two cells of each area per period and for which all cells of the same area are topologically equivalent. For λ close to 1 the minimal tiling is hexagonal. For smaller values of λ the minimal tilings contain pairs of 5/7, 4/8 and 3/9 cells, the cells with fewer sides having smaller area. The correlation between area fraction and number of sides in the minimal tilings is approximately linear and consistent with Lewis' law.

Fast equi-partitioning of rectangular domains using stripe decomposition

This paper presents a fast algorithm that provides optimal or near-optimal solutions to the minimum perimeter problem on a rectangular grid. The minimum perimeter problem is to partition a grid of size M × N into P equal-area regions while minimizing the total perimeter of the regions. The approach taken here is to divide the grid into stripes that can be filled completely with an integer number of regions. This striping method gives rise to a knapsack integer program that can be efficiently solved by existing codes. The solution of the knapsack problem is then used to generate the grid region assignments. An implementation of the algorithm partitioned a 1000 × 1000 grid into 1000 regions to a provably optimal solution in less than one second. With sufficient memory to hold the M × N grid array, extremely large minimum perimeter problems can be solved easily.

Extremal Polygons with Minimal Perimeter

In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variations of the isoperimetric problem for this class of polyhedra. The first one is: Which bodies have minimal surface area in the class of extremal bodies for a fixed n-dimensional lattice? And the second one is: Which bodies have minimal surface area in the class of extremal bodies with volume 1 of dimension n? We characterize the solutions of these two problems in the plane. There is a consequence of these results, the solutions of the above problems in the plane give the solution of the lattice-like covering problem: Determine those centrally symmetric convex bodies whose translated copies (with respect to a fixed lattice L) cover the space and have minimal surface area.

Three-Dimensional Analysis of a Thermal Dissipation System with a Rectangular Diamond Heat Spreader on a Semi-infinite Copper Heat Sink

We consider the 3-D problem of conductive heat dissipation from a rectangular patch source of constant heat flux, through a finite rectangular parallelepiped heat spreader into a semi-infinite heat sink. Using the Fourier analysis, we have developed a rigorous series solution to this 3-D boundary value problem. Considering the case of a diamond heat spreader with a copper heat sink, we have investigated the effects of the geometrical dimensions of the spreader and the shape of the rectangular heat patch source on the thermal behavior of the system. The main results we have obtained are: (1) the commonly-used assumption of isothermal spreader-sink interface is valid provided that the spreader thickness is relatively large compared with the heating-patch's length and width; (2) for a square heat spreader with a fixed area ratio of rectangular heat patch to the spreader's cross section, the maximum temperature occurs for square heat patches as a result of the minimum perimeter; (3) for the same thickness and cross-section area of the spreaders, circular systems can be used to predict an upper bound of thermal behaviors for the rectangular heat dissipation systems.

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

Initial conditions and dynamics of triple systems

The dynamical evolution of triple systems with equal-mass components and zero initial velocities is studied. We consider two regions of initial conditions: a regionD of all possible configurations of triples and a circleR. The configurations are distributed uniformly within these regions. The calculations have been carried out until a time when escape or ‘conditional’ escape (i.e. distant ejection) of one component takes place. The accuracy has been checked by doing time-reversed integration. Types of ‘predictable’ and ‘non-predictable’ systems are revealed. Averages for a number of evolution parameters are presented: the life-time, minimum perimeter during the last triple approach resulting in escape, semi-major axis and eccentricity of the final binary, and the smallest separation between the components during the evolution. It is shown that the statistical results for the regionsD andR do not differ significantly for the most part. Our results, which have been obtaned by a three-body regularization method, are in good agreement with previous work based on the RK4 integrator and Sundman's time smoothing.

Main features of dynamical escape from three-dimensional triple systems

The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper.

The geometry of circumscribing polygons of minimal perimeter

LetC be a closed strictly convex curve in the Euclidean plane, and letC(n) denote the class ofn-sided polygons which circumscribeC. Geometric conditions are given for polygonsP ∈C(n) which have minimum perimeter in the classC(n).

FITOSSOCIOLOGIA DA VEGETAÇÃO ARBÓREA DA RESERVA FLORESTAL DO SACAVÉM, SÃO LUÍS, MARANHÃO (BRASIL)

Foi amostrada fitossociologicamente uma área de 0,5 ha de mata na Reserva Florestal do Sacavém, distando cerca de 7 Km do centro da cidade de São Luís (2"32'S; 44"17'W), em manchas remanescentes de floresta tropical úmida, denominada localmente de "Pré-Amazônia". A área foi dividida em 50 parcelas de l0x 10 m ( 100 m2), nas quais foram amostrados 410 indivíduos com PAP a partir de 15 cm, pertencentes a 34 famílias, 66 gêneros e 110 espécies, além de 5 indivíduos mortos não considerados na análise fitossociológica. O DAP variou de 4,8 cm a 86,7 cm, ficando a média cm 16,28 cm. As alturas mínima, média e máxima foram, respectivamente, 2,0 m, 11,16 m e 25,0 m. A área basal total foi de 14,207 m2, equivalente a 28,4140 m2/ha, com média de 0,07 m2/indivíduo, e a densidade absoluta foi de 820 indivíduos/ha. As cinco famílias mais importantes em IVI foram Leguminosae, Chrysobalanaceae, Meliaceae, Myrtaceae e Arecaceae, perfazendo 49,24% do total. As primeiras cinco espécies em IVI foram Licania cf incana Aubl., Guarea guidonia (L.) Sleumer, Copaifera langsdorffii Desf., Dipteryx laamiferaDucke c uma morfo-espécie (Desconhecida 1). Cerca de 60% das espécies consideradas apresentaram apenas 1 ou 2 indivíduos. O índice de diversidade de Shannon e Weaver (Η') foi de 4,189. Foram feitas comparações com diversos trabalhos realizados em florestas amazônicas, confirmando que a área em estudo é comparável àquelas em riqueza, densidade e dominância, embora a área tenha sido subdimensionada.

ASPECTOS FLORÍSTICOS QUANTITATIVOS E COMPARATIVOS DA VEGETAÇÃO ARBÓREA DA RESERVA FLORESTAL DO SACAVÉM, SÃO LUÍS, MARANHÃO (BRASIL)

Foi inventariado 0.5 ha de mala localizado na Reserva Florestal tio .Sacavém. parte integrante do Parque Estadual do Bacanga. que dista cerea de 7 Km do centro de São Luís (2"32'S: 44"I7'W), capital do Estado do Maranhão, numa altitude inferior a 30 in. A área foi coberta originalmente por uma floresta tropical úmida, denominada localmente de "Pré-Amazônia". O estudo foi realizado nas manchas remanescentes de vegetação. dividido em 50 parcelas de 10x10 m. nas quais foram amostrados 410 indivíduos com PAP (perímetro ã aluíra do peno) a partir de 15 cm. pertencentes a 34 famílas, 66 gêneros e 110 espécies, além de 46 espécies arbóreas, pertencentes a 41 gêneros e 25 famílias, que não foram incluídas na análise quantitativa. O DAP variou de 4.8 cm a 86.7 cm. ficando a média em 16.28 cm. As allur.is mínima, média e máxima foram, respectivamente, 2,0 m, 11,16 m e 25,0 ni. Foram leitas comparações com diversos trabalhos realizados na Amazônia, indicando que a área estudada apresenta uma riqueza elevada, sendo comparável àquelas florestas também em estrutura, embora os valores médios de alluni e diâmetro sejam mais baixos.

For Every Answer There are Two Questions

Duality is a concept that dominates modem optimization theory. The most familiar one is that associated with linear programming. Here we look at a different one. Most students are introduced to optimization in a first course in calculus. We propose to show that the idea and use of duality can be introduced in such a course. In a general way, duality asserts the existence of two solutions of extrema problems that are intimately related. Ideally, the two problems should use the same data; one problem should have a solution if and only if the other one does; and the solution of one should give information about the solution of the other. Consider by way of example two problems that are in every calculus text. I. Find the rectangle of maximum area when the perimeter is fixed. II. Find the rectangle of minimum perimeter when the area is fixed. An alert calculus student will observe that the extremal rectangle in both problems is a square. Is this an accident? One way to begin is to observe that Every time you solve an optimization problem you have proved an This is hardly a deep statement, but it is a useful principle. For example, if one has solved Problem I and discovered that the answer is a square then one can formulate a useful inequality. For let A be the area of any rectangle of perimeter P. Then A <A1 where A1 is the area of the square of perimeter P. But A1 = (P/4)2 so that

Finding minimum areak-gons

Given a setP ofn points in the plane and a numberk, we want to find a polygon[Figure not available: see fulltext.] with vertices inP of minimum area that satisfies one of the following properties: (1)[Figure not available: see fulltext.] is a convexk-gon, (2)[Figure not available: see fulltext.] is an empty convexk-gon, or (3)[Figure not available: see fulltext.] is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn3). The space complexity isO(n) fork=4 andO(kn2) fork?5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

Computing simple circuits from a set of line segments

We address the problem of connecting line segments to form the boundary of a simple polygon--a simple circuit. However, not every set of segments can be so connected. We present anO(n logn)-time algorithm to determine whether a set of segments, constrained so that each segment has at least one endpoint on the boundary of the convex hull of the segments, admits a simple circuit. Furthermore, this technique can also be used to compute a simple circuit of minimum perimeter, or a simple circuit that bounds the minimum area, with no increase in computational complexity.

Reproductive Activity in an Old-Age Coyote in Southeastern Colorado

Old ages reported for wild coyotes (Canis latrans) were 13.5 years old for a male (Knowlton, 1972; Nellis and Keith, 1976) and 14.5 years old for a female (Knowlton, 1972). No information was available on the past history of these wild coyotes. I documented a 15.4-year-old male coyote in southeastern Colorado which, on the basis of behavioral activity, appeared to be reproductively active when 11.6 to 12.8 years of age. On 9 November 1983, a 12.0-kg male coyote was captured with a steel leg-hold trap in Lockwood Canyon on the U.S. Army Pifion Canyon Maneuver Site, Las Animas Co., Colorado. Age at capture was estimated by tooth wear (Gier, 1968) to be > 8 years. From 10 November 1983 to 8 November 1985, a total of 182 radio locations of the coyote were collected. The last day of radio contact was 8 November 1985. Homerange size, as estimated by the minimum perimeter polygon method (Mohr, 1947), was 7.7 and 8.2 km2 for 1984 and 1985, respectively. On 21 September 1987, the coyote was shot by a rancher 7.7 km outside the boundary of its known home range. The carcass was recovered; the lower jaw was boiled for 1 h to remove a canine tooth which was sent to a commercial toothprocessing laboratory (Matson's Laboratory, Milltown, Montana) for sectioning and cementum analysis (Linhart and Knowlton, 1967). Age analysis showed the coyote was 15.4 years old at the time of death. Matson's Laboratory gave the age estimate an accuracy rating of A, their highest rating with an associated error of ? 1 year. While cementum analysis may have some inaccuracies (Allen and Kohn, 1976; Nellis et al., 1978; Roberts, 1978), previously reported ages were also aged by cementum analysis and were thus subject to the same errors.

The Decomposition of a Rectangle into Rectangles of Minimal Perimeter

We solve the problem of decomposing a rectangle R into p rectangles of equal area so that the maximum rectangle perimeter is as small as possible. This work has applications in areas such as flexible object packing and data allocation. Our solution requires only a constant number of arithmetic operations and integer square roots to characterize the decomposition, and linear time to print the decomposition. The discrete analogue of the problem in which the rectangle R is replaced by a rectangular array of lattice points is also considered, and three heuristic methods of solution are given. All of the heuristic methods operate by finding a discrete approximation to our optimal decomposition of R, but with different tradeoffs between the accuracy of the approximation and running time.

The decomposition of a square into rectangles of minimal perimeter

This paper solves the problem of subdividing a unit square into p rectangles of area 1/p in such a way that the maximal perimeter of a rectangle is as small as possible. The correctness of the solution is proved using the well-known theorems of Menger and Dilworth.

Scan-Along Polygonal Approximation for Data Compression of Electrocardiograms

Three fast and efficient scan-along algorithms for compressing digitized electrocardiographic data are described. These algorithms are scan-along in the sense that they produce the compressed data in real time as the electrocardiogram is generated. The algorithms are based on the minimum perimeter polygonal approximation for digitized curves. The approximation restricts the maximum error to be no greater than a specified value. Our algorithms achieve a compression ratio of ten on a database of 8000 5-beat abnormal electrocardiograms sampled at 250 Hz and a compression ratio of eleven on a database of 600 3-beat normal electrocardiograms (different from the preceding database) sampled at 500 Hz.

Scale Preserving Smoothing of Polygons

A smoother version of a polygon ¿ is defined as a polygon which approximates ¿ according to a given criterion and which simultaneously has no more edges than ¿ itself. In this paper, a scale preserving smoothing algorithm is presented. The input to the algorithm is a polygon ¿ and the output is its smoothed version ¿. ¿, which contains all the scale information that ¿ contains, is called the linear minimum perimeter polygon (LMPP) of ¿ within a tolerance of. Using the quantity the degree of with ¿ approximates ¿ can be controlled. From the LMPP a representation for a polygon approximating ¿ can be procured, which is invariant to scale and translation changes. Examples of smoothing maps and characters have been presented.