Showing papers on "Polygon published in 1971"

Continuous Shading of Curved Surfaces

Abstract: A procedure for computing shaded pictures of curved surfaces is presented. The surface is approximated by small polygons in order to solve easily the hidden-parts problem, but the shading of each polygon is computed so that discontinuities of shade are eliminated across the surface and a smooth appearance is obtained. In order to achieve speed efficiency, the technique developed by Watkins is used which makes possible a hardware implementation of this algorithm.

Computer display of curved surfaces

Abstract: : The report describes a method for producing shaded pictures of curved surfaces. It uses a small polygon approximation of the surface to solve efficiently the hidden parts detection, and then computes the shading on each polygon in such a way that visual discontinuities between adjacent polygons disappear, thus restoring the apparent smoothness of the surface and increasing greatly the realism of the pictures produced. The smooth shading technique described here has been used to produce a large variety of pictures of which several airplanes, a car, a human face and some mathematical surfaces are included to illustrate the effect of the method. (Author)

Optimal location of a single service center of certain types

Abstract: Hakimi has considered the problem of finding an optimal location for a single service center, such as a hospital or a police station. He used a graph theoretic model to represent the region being serviced. The communities are represented by the nodes while the road network is represented by the ares of the graph. In his work, the objective is one of minimizing the maximum of the shortest distances between the vertices and the service center. In the present work, the region being serviced is represented by a convex polygon and communities are spread over the entire region. The objective is to minimize the maximum of Euclidian distances between the service center and any point in the polygon. Two methods of solution presented are (i) a geometric method, and (ii) a quadratic programming formulation. Of these, the geometric method is simpler and more efficient. It is seen that for a class of problems, the geometric method is well suited and very efficient while the graph theoretic method, in general, will give only approximate solutions in spite of the increased efforts involved. But, for a different class of problems, the graph theoretic approach will be more appropriate while the geometric method will provide only approximate solutions though with ease. Finally, some feasible applications of importance are outlined and a few meaningful extensions are indicated.

A heuristic to reduce the wrapping effect in the numerical solution ofx′=f(t,x)

Abstract: The problem of obtaining a realistic guaranteeda posteriori bound on the accumulated error in a computed solution to the initial value problem in ordinary differential equations is difficult, because of the “wrapping” effect. This difficulty can sometimes, but not always, be avoided by making use of coordinate transformations. In this paper we propose that the wrapping effect be reduced by enclosing the accumulated error in a convex polygon of a certain form, and we describe one possible way of choosing the faces of such a polygon. The method is computationally expensive, but provides, in cases where other methods are unable to do so, a bound which does not grow exponentially too fast.

A nonstandard proof of the Jordan curve theorem

Abstract: In this paper a proof of the Jordan curve theorem will be presented. Some familiarity with the basic notions of nonstandard analysis is assumed. The rest of the paper is selfcontained except for some standard theorems about polygons. The theorem will be proved in what ought to be a natural way: by approximation by polygons. This method is not usually found in the standard proofs since the approximating sequence of polygons is often unwieldly. But by using nonstandard analysis, one can approximate a Jordan curve by a single polygon that is infinitesimal ly close to the curve. This allows types of reasoning which are extremely difficult and unnatural on sequences of polygons. Preliminaries. The basic concepts of nonstandard analysis and some acquaintance with polygons are assumed. Some basic definitions and theorems of point set topology are also assumed. Throughout this paper the following notations and conventions will be used: (1) All discussion, unless otherwise stated, is assumed to be about a nonstandard model of the Euclidean plane. 'Otherwise stated" will often mean that the notion or concept will be prefaced by the word "standard". (2) A standard concept and its extension will be denoted by the same symbol. If it is necessary to distinguish between them, reference to the model in which they are to be interpreted will be made. (3) If A and B are sets of points and x is a point, then \x, A\ will denote the distance from x to A and \B, A\ = mίxeB\x, A\. (Thus if A Π B Φ 0 then \A, B\ = 0.) \x,y\ will denote the distance from the point x to the point y. (4) / will denote a fixed continuous function on [0,1] into the Euclidean plane with the property that x < y and f(x) — fyy) if and only if x — 0 and y = 1. C will denote the range of /. ( 5 ) x ~ y will mean that the distance from x to y is infinitesimal. If x is near-standard then °χ will denote the standard y such that x ~ y. (6) \i x and y are points then xy will denote the ordered, closed line segment that begins at x and ends at y. (7) If x and y are points then intv (x, y) is the set of all points z of xy such that z Φ x and z Φ y.

Chess game apparatus

Abstract: Game apparatus comprising a board and player pieces are provided whereby each of three players may simultaneously compete against the other two. The board is in the shape of a six-sided polygon of which three relatively long sides alternate with three relatively short sides. The face of the board is divided into 141 hexagons of three colors, there being 47 hexagons of each color, no two adjacent hexagons being of the same color. Used with the board are three sets of 18 player pieces, each set including 4 pawns and 14 major pieces.

Multiple line rotating polygon

Abstract: An optical scanning device is provided for producing a multiple line scan. A rotating refractive scanning prism in the form of a polygonal solid having an even number of revolving opposite parallel faces provides a multiple line scan, the lines being separated because the plane parallel pairs of revolving opposite faces are at a small angle to the rotating axis of the prism, which rotating axis is generally perpendicular to the optical axis. The number of lines scanned in such an arrangement is equal to the number of revolving faces on the polygon, and can be increased, in another embodiment of the invention, by providing a plurality of detectors in an array to produce a line raster having a number of lines equal to the number of detectors times the number of revolving sides of the polygon.

A method for eliminating hidden lines with polyhedra

Abstract: A method is given to eliminate hidden lines from line drawings of volumes whose surfaces are a union of polygons. Projected polygons are tested pair- wise for overlap by looking for lines with the property that all vertices of one polygon lie on one side of the line and all vertices of the other pdygon lie on the opposite side. When a pair is found to overlap, hidden line endpoints are found by searching a table that was constructed in the test for overlap. Some results of a program written to demonstrate the method are given.

Continuation rates of use with a closed, non-tailed intrauterine device: polygon (M) and Lippes' loop (B).

Abstract: This Indian study which ran 2 years beginning in July 1967 compared the Lippes loop B (119 women) with the polygon (M) 120. Because of the high rate of loop termination during the first year second-year follow-up was done only for polygon wearers. The patients aged 15-50 multiparous and free from the usual contraindications came largely from the rural and semiurban areas of Ballabgarh Community Development Block which is 30 km from New Delhi. At 3 months the data showed that a woman wearing a loop had a 4.5 times greater chance of expulsion or removal for medical reasons than did a woman with a polygon. However most of the differential risks for events occurred in those first 3 months. At 9 months 48.87% and 79.74% of loop and polygon wearers respectively continued use. 65.99% of the polygon women remained active at the end of the second year while only 38.53% of the loop women remained active at the end of the first year. At 1 year no loop wearer had become pregnant but at 2 years 10.31% of the polygon patients had become pregnant. Thus the polygon showed a failure rate comparable to that of other closed devices a rate much higher than the ones characteristic of open devices. Since the polygon has no tail it was necessary for a woman to contact a doctor when wishing removal and the more frequent doctor-patient sessions helped the continuation rate. The shape of the polygon was a major factor contributing to the infrequency of expulsions. To lessen the pregnancy rate and the perforation risk the polygon could make use of copper or zinc and fine collapsible membrane.

A semi-submersible floating structure

Abstract: 1,221,871 Floating offshore structure; boring earth &c INSTITUT FRANCAIS DU PETROLE DES CARBURANTS ET LUBRIFIANTS 6 March, 1968 [6 March, 1967], No 10939/68 Headings E1F and E1H A semi-submersible floating structure for offshore drilling operations includes streamlined caissons such as 5, 6, 7 located at the a pices of a regular polygon having at least three sides, the caissons having a common direction of orientation parallel to an axis of symmetry XX' of the polygon and being interconnected by cross-bracing members 8, 9, 10 Columns 2, 3, 4 integral with the caissons support a platform 1 which is a hollow structure having an acrodynamic flattened shape which is substantially a surface of revolution around vertical axis ZZ' In the structure shown platform 1 is an ellipsoid of revolution the small axis of which is vertical and the regular polygon is an equilateral triangle Caissons 5, 6, 7 are provided with units 12, 13, 14 respectively for propulsion in their longitudinal direction and with units 15, 16, 17 respectively for transverse propulsion so that the structure may be dynamically anchored Derrick 41 is located on the vertical axis ZZ' of platform 1 and the platform includes a rotation table and an annular plate 43 rotated by motor 45 Columns 2, 3, 4 may be used for stacking the elements of a drill string, Fig 4 (not shown)

frequency distributions and graphs

Abstract: The primary goal of descriptive statistics is to bring order out of chaos. Descriptive statistics help resolve problems by making it possible to summarize and describe large quantities of data. Among the various techniques that are found particularly useful are the following: (1) frequency distributions and graphs; (2) measures of central tendency; (3) measures of variability; and (4) transformed scores. Each of these procedures serves a different and important function. The first step in constructing a regular frequency distribution is to list every score value in the first column. This chapter explains how to make a complete regular frequency distribution table, a cumulative frequency distribution table, and grouped frequency distributions. It is often effective to express frequency distributions pictorially as well as in tables. The two procedures for accomplishing this are: (1) histograms or bar graphs and; (2) regular frequency polygons and cumulative frequency polygons. Cumulative frequency distributions are also commonly graphed in the form of frequency polygons, and the resulting figure is called a cumulative frequency polygon. The chapter also discusses the shapes of frequency distributions.