Showing papers on "Polygon published in 1976"

Location of a point in a planar subdivision and its applications

Abstract: Given a subdivision of the plane induced by a planar graph with n vertices, in this paper we consider the problem of identifying which region of the subdivision contains a given test point. We present a search algorithm, called point-location algorithm, which operates on a suitably preprocessed data structure. The search runs in time at most 0((log n)2), while the preprocessing task runs in time at most 0(n log n) and requires 0(n) storage. The methods are quite general, since an arbitrary subdivision can be transformed in time at most 0(n log n) into one to which the preprocessing procedure is applicable. This solution of the point-location problem yields interesting and efficient solutions of other geometric problems, such as spatial convex inclusion and inclusion in an arbitrary polygon.

On convex lattice polygons

Abstract: Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.

Parallel Detection of Concavities in Cellular Blobs

Abstract: This paper reports some results on the use of parallel-structured computers to detect and describe concavities in simply connected planar regions (``domains'' or ``blobs''). We show, in particular, how these concavities may be obtained by a parallel filling-in process-somewhat like pouring liquid into several cups simultaneously. It has been shown that the concavities and concavity tree of a regular cellular blob (i.e., a digitized simply connected planar region) can be obtained by the use of a sequential algorithm that finds the minimum-perimeter polygon (MPP) passing through the boundary cells of the cellular blob. In this paper we show how any such MPP may be computed by a sequence of simultaneous local operations in a parallel-structured computer. We also show that the ratio of computation times for sequential algorithms to those for parallel algorithms operating on the cellular image of a large circular blob is approximately proportional to the square root of the blob's perimeter, assuming the size of the vertex-detecting window is fixed and large enough to detect one or more vertices of the MPP. We also show that filling in the concavities by a sequence of parallel local operations terminates to an approximation of the cellular hull (i.e., the digitization of the convex hull of the original blob) in a finite time, and that this approximation is a subset of the cellular hull.

Roof reflective polygon scanning apparatus

Abstract: The present invention provides a multifaceted reflecting polygon scanner system that is suitable for incorporation into a projector, camera, or other optical scanning or image immobilizing apparatus. The rotating reflecting optical system of the present invention, is capable of being combined with a continuous film transport to produce an overlap frame dissolve of sequential frames for optical immobilization of a projected image. The present invention includes a primary roof reflecting polygon, a secondary reflecting polygon, and an image surface. By proper choice of the polygon parameters and their juxtaposition relative to the image surface, the present invention uniquely provides a significantly higher relative aperture with minimal kinetic aberrations such as dynamic keystone distortion and minimal static aberrations. In one embodiment, a pair of roof reflectors of a 60° and 90° included angle prevent the generation of ghost images while providing an aberration free image with an isotransport film motion. When incorporated into a projection system, a projection lens having low geometric distortion and a long back focal length can be utilized wih the optical scanning system. The exact form of the projection lens would depend upon the particular adaptation of the reflecting polygon scanning system.

Some Concepts and Descriptions of Physiological Tolerance: Rate–Temperature Curves of Poikilotherms as Transects of Response Surfaces

Abstract: Precht rate curves may be considered as transects across a rate surface bounded by a Fry temperature polygon. Changes in metabolic or active capacity within the tolerance polygon approximate capaci...

Self-consistent GTD formulation for conducting cylinders with arbitrary convex cross section

Abstract: A user-oriented computer program has been developed for high frequency radiation and scattering from infinitely-long perfectly. conducting convex cylinders. The analysis is based on the self-consistent geometrical theory of diffraction (GTD). The cylinder is modeled as an N -sided polygon. Two cylindrical waves with unknown amplitudes are assumed to travel in opposite directions on each face of the polygon. The boundary conditions for the corners are applied to set up a matrix equation for 2N unknowns (the amplitudes associated with the traveling cylindrical waves). Crout's method is used to solve the matrix equation. Once the amplitudes for the traveling waves are determined, the radiation or scattered field is readily obtained via the usual GTD techniques. Numerical results are presented for radiation and scattering from rectangular, semi-circular, circular, and elliptic cylinders for both principal polarizations. The results show excellent agreement with GTD, moment, and eigenfunction solutions.

Nut cracking machine

Abstract: A nut cracking machine comprising a plurality of wheels rotatable by a power drive on axes forming a closed, planar polygon, the peripheries of the wheels meeting at the center of the polygon, peripheral grooves of arcuate cross-sectional form in the peripheries of the wheels conjointly forming a circular orifice between the wheels in the plane of the polygon, through which nuts to be cracked are passed.

Computer program for presentation of grain-size data by the graphic method

Abstract: A computer program is described for a Hewlett-Packard desk-top Calculator (Model 9820A)-Plotter (Model 9862A) which plots a histogram, frequency polygon and/or cumulative curve, as well as standard statistical parameters and percentages of gravel, sand, silt, and clay from raw weight data from a sediment grain-size analysis. The program utilizes the graphic method for calculating statistical parameters in contrast to other commonly used computer programs derived for the method of moments. This program is advantageous because (1) it is rapid, (2) it allows direct comparison with grain-size data in the literature which have been computed by the graphic method, (3) either a complete presentation of a distribution can be obtained on one sheet of paper or a series of cumulative curves or frequency polygons can be prepared on a single graph which is ready for publication photocopying, (4) automatic extrapolation of data points from a cumulative curve eliminates inconsistencies arising from manual extrapolation, and (5) the desk-top calculator is more convenient to use, eliminates punching of data cards, and can be operated by someone without a knowledge of computers and programming. Disadvantages of the program are: (1) some authors feel critical percentiles are more accurately read off cumulative curves plotted on a probability ordinate than on an arithmetic ordinate, (2) the cumulative curve is a series of straight-line segments, (3) the frequency polygon is not as accurate a representation of grain-size distribution as a frequency curve and (4) size class intervals for the histogram and frequency polygon are standardized at 0.5φ, even though gravel and mud fractions are analysed at whole φ intervals. For most studies, the advantages of the program outweigh the disadvantages.

On the number of sides necessary for polygonal approximation of black-and-white figures in a plane

Abstract: A bound on the number of extreme points or sides necessary to approximate a convex planar figure by an enclosing polygon is described. This number is found to be proportional to the fourth root of the figure's area divided by the square of a maximum Euclidean distance approximation parameter. An extension of this bound, preserving its fourth root quality, is made to general planar figures. This is done by decomposing the general figure into nearly convex sets defined by inflection points, cusps, and multiple windings. A procedure for performing actual encoding of this type is described. Comparisons of parsimony are made with contemporary figure encoding schemes.

A new Clitocybe from the Alaskan Arctic

Abstract: A new species of Clitocybe is described from the Alaskan Arctic. This agaric grows on low-center polygon rims, high-center polygons, and on beach ridges. Drawings of microscopic features and macrop...

Hexagonally shaped rotating display with tetrahedral elements

Abstract: An improved mobile display apparatus is provided for advertising, decorative, or other purposes, which is made up of a series of solid geometrical polygons linked to one another by hinges or universal joints, such that the movement of one polygon is transmitted to the others. The polygons may be tetrahedrons or other four-sided solid geometrical elements whose triangular sides have edges of unequal length. An even number of six or more tetrahedrons, for example, may be joined together in accordance with the invention in a continuous configuration. Odd or even numbers of three-dimensional polygons constructed from triangular elements with edges of unequal lengths may also be linked to form continuous chains, and folded together to form a display apparatus incorporating the concepts of the invention.

Venous catheter fastening element of flat flexible material - adheres to skin after removal of sterilized outer protective strips

Abstract: The flat flexible material is cut to the shape of a regular mean polygon (1) which has a hole (9) in the centre connected to one side (23) by a slit (8). Each corner (10, 11, 12) of the polygon has an external polygon (2, 3, 4) pivoted to it of similar size and arranged so that they all fit over the centre (1) forming an enclosed polygonal surface. Pref. the side of the centre polygon (1) facing skin has a detachable adhesive protective and sterilising film on it. Pref. the polygons are all equilateral triangles. This disposable fastening element is inexpensive to manufacture and alleviates necessity of large number of adhesive plasters and gauze pads to fix the catheter in position.

Polygon scanner with an annular lip

Abstract: A scanning polygon assembly is provided which comprises a plurality of identical facet blocks arranged in abutting relationship on a platform. The platform has an annular lip against which two corners of each block abut so the facets form chords along the lip to form a polygon. The abutting relationship between the blocks is a tight fit but not a forced fit in order that the blocks are free enough to allow the corners to engage the lip (and thus assure such engagement) due to the centrifugal force exerted thereon during rotation of the polygon.

Focus path of a laser beam deflected by a prismatic polygon mirror: its calculation and optimization

Abstract: The focus path of a laser beam deflected by a prismatic polygon mirror was calculated. It is shown that the larger the focal distance of the objective or the polygon radius, the closer the focus path approaches a straight scanning line. Especially in the case of small pattern scanning, a remarkably small deviation between actual focus path and straight scanning line can be achieved by suitable sizing and arrangement of the optical elements.

Conductor bundle flexible connector - has intersections of individual conductors with orthogonal planes in polygonal configuration

Abstract: The flexible connector has at least four partial conductors combined into a cable bundle in parallel and spaced arrangement. the conductor ends are connected at their ends by a clamp each. The intersection points of the individual conductors (6, 6', 7, 7', 8, 8') with planes through which the individual conductors pass vertically, form a polygonal line configuration with angles exceeding 90 deg. between two adjacent sides. The corners of the polygon are located on a circle or a similar curve. Preferably the sides of the polygon are almost of identical length. The polygons of different planes may be identical. Typically the cable bundle has not more than six individual conductors.

The Algebra and Geometry of Polyhedra.

Abstract: THE success that analytic geometry has had in reducing many geometric questions to algebraic ones sometimes gives the im pression that we can ignore the geometry and concentrate on the algebra. In fact the two areas are complementary and together yield much deeper and more elegant results than either does separately. This has been commented on before (for example, see Farrell and Ranucci 1973). The cardboard and-rubber-band method of constructing polyhedra described so beautifully by B. Stewart in Adventures among the Toroids (1970, p. 20) is of inestimable value in the classroom. It enables the students to experi ment and discover for themselves all the possibilities described here. Some advance preparation is required, however. A large supply of triangles and squares of uniform size (2 inches on a side is convenient and economical) should be constructed before hand by the students, and a sufficient num ber of rubber bands (lv4" long, available in office supply stores as number 10) should be obtained. The topic in this article could be done immediately after a discussion of Euler's formula, and the analysis presented here is much easier than that involved in the investigation of deltahedra (polyhedra all of whose faces are equilateral triangles) described in Beck, Bleicher, and Crowe, (1969). In this article I would like to give a further example in three-dimensional geometry that is easily done in the class room. Imagine that we have a box full of triangles and a box full of squares and we want to build a convex polyhedron with them. We insist that our polyhedron use some of each kind of polygon and that every vertex have the same number and arrangement of these polygons. If we de cide to use m triangles and squares at each vertex then we immediately have the in equality.

A Ruin Problem

Abstract: In the classical ruin problem two players with finite initial capitals play until one of them is ruined. It is often useful in analyzing this problem, for instance in finding the probability that a specific player is ruined and the expected duration of the game, to interpret the game as a random walk on the line with absorbing barriers. We will here treat a ruin problem which can be interpreted as a random walk on a polygon. We start with two simple cases.

The XY-model and the self-avoiding walk approximation

Abstract: The SAW-approximation is calculated for the quantum-mechanical XY-model for spin 1 2 in a parallel magnetic field. For this purpose we derive an exact expression for the internal weight factors of the polygon graphs. In this approximation we obtain an analytic expression for the phase-separation line in the (H, T)-phase. The critical exponents are the same as for classical models.