Orthogonal Polygons Research Articles

Stationing guards in rectilinear art galleries

A rectilinear polygon can be viewed as an art gallery room whose walls meet at right angles. An algorithm is presented that stations guards in such a room so that every interior point is visible to some guard. The algorithm partitions the polygon into L-shaped pieces, a subclass of star-shaped pieces, and locates one guard within each kernel. The algorithm runs in O ( n log n ) time in the worst case for a polygon of n vertices.

On the X-Y convex hull of a set of X-Y polygons

We study the class of rectilinear polygons, calledX – Y polygons, with horizontal and vertical edges, which are frequently used as building blocks for very large-scale integrated (VLSI) circuit layout and wiring. In the paper we introduce the notion of convexity within the class ofX – Y polygons and present efficient algorithms for computing theX – Y convex hulls of anX – Y polygon and of a set ofX – Y polygons under various conditions. Unlike convex hulls in the Euclidean plane, theX – Y convex hull of a set ofX – Y polygons may not exist. The condition under which theX – Y convex hull exists is given and an algorithm for testing if the given set ofX – Y polygons satisfies the condition is also presented.

Orthogonale Erzeugendenpolygone auf einschaligen Hyperboloiden

The question whether there exist closed orthogonal polygons formed by generators of a given hyperboloid is a poristic problem: For a prescribed number 2n of sides there exists either no polygon of the required kind or an infinite continuous set of such polygons. After having led back the solution to the classic theorem of Poncelet, closure conditions are developed for the casesn=2, 3, 4, 5 and 6.

Plastic merge procedure for mosaics of polygons

Graphic input, while providing a congenial medium for the user, can lead to problems of noisy data. In the case of mosaics of arbitrary polygons input graphically, the problem lies in boundary misalignments and the consequent lack of guaranteed common boundaries. A general plastic merge procedure operating on mosaics of arbitrary rectilinear polygons is described which corrects such boundary misalignments. The work was originally motivated by the needs of an applications project funded by sponsors with user interests. With further support from SRC, the underlying concepts were subsequently expanded and are presented here in a form which is not application-specific.

On ordinary points in arrangements

We show that in an arrangement ofn curves in the plane (or on the sphere) there are at leastn/2 points where precisely 2 curves cross (ordinary points). Furthermore there are at least (4/3)n triangular regions in the complex determined by the arrangement. Triangular regions and ordinary vertices are both connected with boundary vertices of certain distinguished subcomplexes. By analogy with rectilinear planar polygons we distinguish concave and convex vertices of these subcomplexes. Our lower bounds arise from lower bounds for convex vertices in the distinguished subcomplexes.

Theory of Brittle Fracture Applied to Soil Cement

The cracking in a soil-cement base is examined using Griffith’s theory of brittle fracture. Flexural tests are performed on soil-cement beams with crack-simulating notches to determine the critical strain energy-release-rate, \IG\dc\N, associated with the rapid extension of the crack. Model studies are made to evaluate the crack propagation in a soil-cement pavement base. The variations of critical strain energy release-rate according to soil texture, clay mineral present, and the time rate of loading, as well as its independence of the notch geometry indicate that \IG\dc\N is a physical property of the cement-treated soil. Fracture toughnes, \IK\dc\N, a function of the stress intensity factor \iK and the crack extension force \iG, in accord with fracture mechanics concepts, exhibits an inverse relation with the rate of crack propagation. In addition, the crack patterns observed in the model slabs are in keeping with the fundamental postulate of Griffith. In simple and symmetrical cases, the cement base upon cracking tends to form a pattern of hexagons; but when the material is more brittle, the cracks form rather poorly defined “irregular random orthogonal polygons”, with the various cracks meeting primarily in three-way nodes.

Giant Desiccation Polygons of Great Basin Playas

Research Article| January 01, 1968 Giant Desiccation Polygons of Great Basin Playas JAMES T NEAL; JAMES T NEAL Terrestrial Sciences Laboratory, Air Force Cambridge Research Laboratories, Bedford, Massachusetts Search for other works by this author on: GSW Google Scholar ARTHUR M LANGER; ARTHUR M LANGER Department of Community Medicine, Mount Sinai Hospital, New York, New York Search for other works by this author on: GSW Google Scholar PAUL F KERR PAUL F KERR Department of Geology, Columbia University, New York, New York Search for other works by this author on: GSW Google Scholar GSA Bulletin (1968) 79 (1): 69–90. https://doi.org/10.1130/0016-7606(1968)79[69:GDPOGB]2.0.CO;2 Article history received: 07 Jul 1966 rev-recd: 05 Jan 1967 first online: 02 Mar 2017 Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation JAMES T NEAL, ARTHUR M LANGER, PAUL F KERR; Giant Desiccation Polygons of Great Basin Playas. GSA Bulletin 1968;; 79 (1): 69–90. doi: https://doi.org/10.1130/0016-7606(1968)79[69:GDPOGB]2.0.CO;2 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietyGSA Bulletin Search Advanced Search Abstract Several previous investigators have recognized giant polygonal fissure patterns in 6 playas of the Basin and Range Physiographic Province (Great Basin). This paper extends the study to 39 playas in Oregon, Nevada, California, Arizona, and New Mexico, with observations on the physical and mineralogical features of fissured and nonfissured areas.Fissures are often 5 m deep and the polygons may attain a width of 300 m. These giant mud cracks develop in clay playas and are attributed to desiccation phenomena. As dehydration proceeds from the surface downward and penetrates the capillary fringe above the water table, shrinkage occurs, which ultimately results in rupture at depth that extends upward to the surface.The mineral constituents of sediments in both the fissured and nonfissured areas are predominantly clay minerals, carbonates, salines, and analcite, with fine grains of quartz, feldspar, and ferro-magnesian silicates. Fissured playas possess significantly greater quantities of clay and carbonate minerals as compared to nonfissured playas. The clay minerals, carbonates, and analcite are primarily present in <2μ size fraction. This colloidal aggregate is believed to exert a major influence on the physical behavior of the sediments which contain the giant polygons. In particular, the dehydration to an almost dry condition of a clay mass, in which the water content may exceed the mineral content, results in a major loss of volume. The shrinkage leads to rupture with the formation of fissures. The fissures form orthogonal polygons characteristic of volume change in a largely uniform horizontal mass with one surface exposed. This content is PDF only. Please click on the PDF icon to access. First Page Preview Close Modal You do not have access to this content, please speak to your institutional administrator if you feel you should have access.

Green's functions for thin perforated elastic slabs

An exact expression in finite terms is found for the small deflexion at any point of an infinitely large plate clamped along an inner curvilinear edge, with outer edge free, and loaded by a concentrated force at an arbitrary point of the plate. The plate can be mapped on the area outside the unit circle by a rational mapping function involving two parameters. By varying these parameters holes having various shapes and several axes of symmetry are obtained. Infinite plates with holes in the forms of regular and approximately rectilinear polygons are included as special cases.

Torsion of beams of ⊥- and ∟- cross-sections

The problem of the torsion of beams of ⊥- and L-cross-sections has received attention from very few authors despite its important technical applications. The first mathematical solution in this connection was obtained by F. Kötter in 1908 for an L-section both of whose arms are infinite. He attacked the problem by the use of the known solution of the rectangle and by application of the scheme of conformal transformation. Kötter's method, however, does not lend itself readily to the solution of the problem involving more than one re-entrant angle. The first solution for the torsion of a beam whose cross-section is a rectilinear polygon of n sides was published in 1921 by E. Trefftz who also applied his method to an infinite L-section. Recently I. S. Sokolnikoff has suggested a more general method depending upon the fundamental theorem of potential theory that a harmonic function is uniquely determined by the values assigned along the boundary of the region within which the harmonic function is sought, the boundary condition and the region being subject to certain well-known assumptions of continuity, connectivity, etc. As an illustration of his method he has given an approximate solution for a ⊥-section whose flange and web are both infinite.

On a solution of Laplace’s equation with an application to the torsion problem for a polygon with reentrant angles

In this paper will be found a general method of solution of the two-dimensional Laplace equation with certain boundary conditions prescribed along the sides of any rectilinear polygon. The applicability of this method to the solution of technical problems will be illustrated by the treatment of the torsion problem for an infinite T-section. The mathematical solution of this problem, and of that for a finite T, is not to be found in the existing literature. I It will be seen that this scheme of solving Laplace's equation can readily be applied to any region which can be mapped conformally onto the upper half-plane in such a way that the boundary of the region goes into the entire real axis while the interior of the region transforms into the upper halfplane. Despite the considerable scientific interest in the behavior of structural members subjected to pure torsion, only a limited number of torsion problems have been brought within the range of mathematical analysis. A torsion problem is solved when one has determined a function 4) which satisfies the equation V2( =0, and which on the boundary of the section subjected to torsion reduces to? 4)* 4(X2 + y2). The determination of 4) for such simple regular sections as the circle, ellipse, equilateral triangle, and rectangle has been achieved with comparative ease.l1 In 1908, F. K6tter? succeeded in obtaining a solution of the torsion problem for an L-section by the use of the known solution for the rectangle and by application of the scheme of conformal transformation. K6tter's

Relation Between the Zeros of a Rational Integral Function and Its Derivate

The theorem: If f'(z) is the first derivate of a rational integral function f(z), then on the Gaussian plane of complex numbers all the roots of the equation f'(z) = 0 lie inside the smallest convex rectilinear polygon surrounding all the roots of the equation f(z) = 0, is well known.* All the proofs for this algebraical theorem are given by using dynamical conceptions, as far as I know.t So I shall prove it here without using any idea of force, but still by starting from the equation