Showing papers on "Polygon published in 1975"

Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields

Abstract: The standard far-field approximation to the Kirchhoff formula for the field scattered by a flat metallic plate S of arbitrary shape is given by a certain surface (double) integral. This double integral can be reduced to a line integral evaluated around the boundary of S. Moreover, if S is a polygon, this line integral can be reduced to a closed form expression involving no integrations at all. The use of such line integral representations can easily reduce the costs of numerical calculation by orders of magnitude. If the integrands are to be sampled p times per wavelength to achieve an acceptable degree of precision, and if A is the area of S , then the numerical evaluation of the double integral requires p^{2}A/\lambda^{2} functional evaluations whereas the line integral only requires p\sqrt{A/\lambda^{2}} . If S is a polygon with N vertices, then only 2N functional evaluations are required to evaluate the closed form expression with no quadrature error at all.

Topological transitivity of billiards in polygons

Abstract: Consider a billiard in a polygon Q⊂R2 having all angles commensurate with π. For the majority of initial directions, density of every infinite semitrajectory in configuration space is proved. Also proved is the typicality of polygons for which some billiard trajectory is dense in phase space.

Computer Analysis of Moving Polygonal Images

Abstract: A general mathematical model is developed as an idealization of the problem of determining cloud motions from satellite pictures. The model consists of superimposed planes of rigid moving polygons. The problem is to determine from a sequence of scenes the linear and angular velocities of the figures, and to decompose the scene into its component figures. Study of the model reveals a number of fundamental relations that form the basis for an analysis program. In particular, a systematic anaylsis is given of the topological changes that can occur when overlapping figures move together or apart. A computer program based on these results is described, and experimental results are presented.

Flying spot scanner with runout correction

Abstract: A flying spot scanning system is provided by utilizing reflected light from a multifaceted rotating polygon which is then directed to the scanned medium. A light source illuminates a portion of the mirrored sides of the polygon during each scanning cycle, to provide a desired sequence of spot scanning. In each scanning cycle, information is transmitted to the scanned medium by modulating the light from the light source in accordance with a video signal. An optical convolution of elements is selected in combination with the light source to assure a uniform spot size throughout the scan width on the scanned medium. One of these elements is a cylindrical lens placed in the optical path between the polygon and the scanned medium for allowing a wide variance in runout tolerance of the scanning system.

Scanning system and method

Abstract: A scanning system and method for reading coded indicia carried on containers, etc., the system and method including means for generating in a predetermined sequence a first light pattern, and a second light pattern, and means for focusing same at a window over which containers carrying the coded indicia are positioned. The disclosure also includes a mirrored rotatable polygon structure including a plurality of mirrors, some of which are inclined at a positive angle and the others which are positioned at a negative angle with respect to the polygon axis of rotation as well as means for adjusting the incline of the mirrors.

Flying spot scanner with plural lens correction

Abstract: A flying spot scanning system is provided by utilizing reflected light from a multifaceted rotating polygon which is then directed to the scanned medium. A light source illuminates a portion of the mirrored sides of the polygon during each scanning cycle, to provide a desired sequence of spot scanning. In each scanning cycle, information is transmitted to the scanned medium by modulating the light from the light source in accordance with a video signal. An optical convolution of elements including at least two lenses having power in one optical plane and substantially no power in the other plane, is provided in combination with the polygon. One of these lenses is placed in the optical path between the polygon and the scanned medium with its power plane substantially parallel to the axis of rotation of the polygon for allowing a wide variance in runout tolerance of the scanning system. The other lens is placed in the optical path between the light source and the polygon with its power plane substantially perpendicular to the axis of rotation of the polygon.

Ergodic properties of a particle moving elastically inside a polygon

Abstract: The flow of a classical particle bouncing elastically inside an arbitrary polygon is investigated. If every interior angle is a rational multiple of π, there exists precisely one isolating integral in addition to the energy; this integral is described in detail; any possible third integral is nonisolating. If one or more interior angles is an irrational multiple of π, the second integral becomes everywhere nonisolating and non‐Lebesgue‐measurable, i.e., the second integral disappears. The flow of two hard points bouncing elastically in a finite one‐dimensional box is equivalent to the flow of a point particle moving elastically inside a right triangle having interior angle tan−1 (m2/m1)1/2, so the preceding remarks apply to this model. Nonrigorous arguments are given in support of the notion that the polygon model is ergodic and mixing, but is not a C‐system.

Total knee-joint prosthesis - has parts forming linkage polygon to give motion resembling motion of natural knee throughout full range of bending

Abstract: Total knee-joint prosthesis has rigid, positively connected operating elements (levers, connector plates, toothed segments etc) to form a linkage polygon which is under load during the whole course of motion and is connected to the femur or tibia by webs and coupling pieces Pref the crossing arms of the polygon are remodelled bodily and jointed to the web and coupling which have rolling surfaces whose generatrices are the polar curves for natural motion of the knee joint and which are connected tangentially to the their surface and positively pref by teeth The motion properties are the same as for a natural knee enabling the patient to walk naturally

Scanning polygon with adjustable mirrors

Abstract: A scanning system and method for reading coded indicia carried on containers, etc., the system and method including means for generating in a predetermined sequence a first light pattern, and a second light pattern, and means for focusing same at a window over which containers carrying the coded indicia are positioned. The disclosure also includes a mirrored rotatable polygon structure including a plurality of mirrors, some of which are inclined at a positive angle and the others which are positioned at a negative angle with respect to the polygon axis of rotation as well as means for adjusting the incline of the mirrors.

Euler-Verfahren für neutrale Funktional-Differentialgleichungen

Abstract: Euler's explicit and implicit polygonal methods are applied to an initial value problem for a neutral functional-differential equation. It is shown that the approximating polygon converges uniformly to the Lipschitz-continuous exact solution.

Container for liquids

Abstract: A container for liquids which takes the form of a spatial polygon having side planes of identical congruent triangles. The container consists of six triangular planes three of which have their bases against the bases of the other three and the top angle of each three meet at a point opposite to the point where the top angle of the other three meet.

Constructional toy with elements of regular polygons - are joined together by ball and socket joints along edges

Abstract: The elements of the construction toy are made of hard plastic in the form of regular polygons, triangles, squares hexagons, etc. in which the length of the side of the triangle equals the length of the side of the square, etc. The square illustrates the general principle in which each side of the polygon has a projecting ball and a socket. The edges of two polygons, e.g. a square and a hexagon, can be fitted together, the ball of each member fitting into the socket of the other so that a hinge is formed. Further polygons can be assembled to form plane surface patterns or three dimensional structures.

Drive shaft coupling of rubber - polygonal sided ring with transverse shaft connections has recesses to aid heat dissipation

Abstract: Drive shaft couplings of rubber is of polygonal plan-form and has metal tubes bonded into the rubber at each of the apices formed between adjacent sides. The axes of the tubes are transverse to the coupling to enable alternate bolted connections to the driving and driven shafts. To avoid the excessive overheating of the rubber during the deflections associated with torque transmission each side of the polygon has elongate recesses to aid heat dissipation from the core. Preferably, each side of the polygon is of square section with a square section formed recess. Recesses may be machined but are preferably moulded into the coupling.

Hypergraphs and abstract polygons

Abstract: The notion of a graph has recently been generalized to include structures called hypergraphs which have two or more vertices per edge. A hypergraph is called 2-settled if each pair of distinct vertices is contained in at most one edge. A connected 2-settled hypergraph which has at least two edges through each vertex might be called an abstract polygon. Lemma: Every abstract polygon contains a cycle. Shephard and Coxeter have examined certain abstract polygons called regular complex polygons, each of which is denoted by a symbol p {q} r where there are p vertices on each edge and r edges through each vertex. Theorem: The girth of the non-starry regular complex polygon p {q} r is q. Thus, the number q is finally given a simple combinatoric interpretation.

Drive sprockets for vehicle tracks

Abstract: An endless track for a tracked vehicle is described The track is driven by an arrangement which includes polygonal drive sprockets Each sprocket has teeth at the corners of the polygon and outwardly curved surface portions between the teeth

Two-way tube post system - has tubes connected to polygonal junction corners via points

Abstract: The system incorporates a number of tubes operating in both directions, each connected to a different corner of a polygon junction, the number of whose corners is the same as the number of tubes. Points are provided at each corner to direct the slugs down one of two sections of the polygon, the sections between the points acting as buffer sections. Each buffer section can have an outlet to a atmosphere from the middle, dividing it into two buffer stations, at the beginning of each of which is contact actuated by the slug. A valve-controlled outlet to atmosphere c an be provided for each tube before it joins the junction.

Computer Simulation of Bubble Motion

Abstract: A dynamic calculation of a magnetic bubble domain which is rigorous within the scheme ofthe domain wall theory is carried out. The bubble base is approximated by a polygon. When bubble deformation is described with appropriate coordinates, the equations of wall motion representing dynamic force balance on the vertices of the polygon are reduced to a set of in-dependent equations with the degree of freedom equal to the number of the vertices. The stability of the solution is improved considerably by taking the effect of the movement of the neighbour vertices into consideration when the wall force is evaluated. The principle of the calculation and results showing the validity of the calculation are given.

Socket joint for fishing rods - sprung piece made of two hollow sections of different cross-section fitting into one another

Abstract: The two sections being fitted together may have a different cross-section, such as round one and a multi-sided one, with or without rounded corners. The sides may be crescent-shaped. The sprung pieces may be straight, spiral, continuous and in parts mis-shapen, or all cross-sectional shapes independent of the direction of twist to the lengthwise axis of the socket-joint. Various combinations of geometrical cross sections can be used, e.g. polygon within a circle, square within a circle, distorted circle within a polygon circle within a periodically distorted circle etc.

Computer Animation Using Parametric Models

Abstract: Computer animation can be implemented using parametric polygonal models. A parametric model consists of a set of polygons, a set of functions and a set of parameters. The position of each polygon vertex is determined as a function of one or more of the para-meters. With parametric models, animation is reduced to specifying the parameter values for each frame of a sequence. Two models are used to illustrate this concept.© (1975) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.