Showing papers on "Polygon published in 1977"
20 Jul 1977
TL;DR: In this paper, a polygon hidden surface and hidden line removal algorithm is presented, which recursively subdivides the image into polygon shaped windows until the depth order within the window is found.
Abstract: A polygon hidden surface and hidden line removal algorithm is presented. The algorithm recursively subdivides the image into polygon shaped windows until the depth order within the window is found. Accuracy of the input data is preserved.The approach is based on a two-dimensional polygon clipper which is sufficiently general to clip a concave polygon with holes to the borders of a concave polygon with holes.A major advantage of the algorithm is that the polygon form of the output is the same as the polygon form of the input. This allows entering previously calculated images to the system for further processing. Shadow casting may then be performed by first producing a hidden surface removed view from the vantage point of the light source and then resubmitting these tagged polygons for hidden surface removal from the position of the observer. Planar surface detail also becomes easy to represent without increasing the complexity of the hidden surface problem. Translucency is also possible.Calculation times are primarily related to the visible complexity of the final image, but can range from a linear to an exponential relationship with the number of input polygons depending on the particular environment portrayed. To avoid excessive computation time, the implementation uses a screen area subdivision preprocessor to create several windows, each containing a specified number of polygons. The hidden surface algorithm is applied to each of these windows separately. This technique avoids the difficulties of subdividing by screen area down to the screen resolution level while maintaining the advantages of the polygon area sort method.
301 citations
TL;DR: A search algorithm, called point-location algorithm, which operates on a suitably preprocessed data structure, which yields interesting and efficient solutions of other geometric problems, such as spatial convex inclusion and inclusion in an arbitrary polygon.
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 $O((\log n)^{2})$, while the preprocessing task runs in time at most $O(n \log n)$ and requires $O(n)$ storage. The methods are quite general, since an arbitrary subdivision can be transformed in time at most $O(n \log n)$ 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.
220 citations
TL;DR: A polygon hidden surface and hidden line removal algorithm is presented that recursively subdivides the image into polygon shaped windows until the depth order within the window is found.
Abstract: A polygon hidden surface and hidden line removal algorithm is presented. The algorithm recursively subdivides the image into polygon shaped windows until the depth order within the window is found....
142 citations
TL;DR: A fast algorithm for testing the inclusion of a point in a many-sided polygon is given and it is shown that the points of intersection of two polygonal lines can be located by what is essentially a binary tree search.
Abstract: A representation for polygons and polygonal lines is described which allows sets of consecutive sides to be collectively examined. The set of sides are arranged in a binary tree hierarchy by inclusion. A fast algorithm for testing the inclusion of a point in a many-sided polygon is given. The speed of the algorithm is discussed for both ideal and practical examples. It is shown that the points of intersection of two polygonal lines can be located by what is essentially a binary tree search. The algorithm and a practical example are discussed. The representation overcomes many of the disadvantages associated with the various fixed-grid methods for representing curves and regions.
43 citations
TL;DR: In this article, it was shown that T is a bounded operator on L"(R2), 1 a > 0, for every p between 4/(3 + 2a) and 4/(1 - 2a).
Abstract: Let us denote by P a polygon of N sides in R2 and consider the operator defined by Tf(e) = Xp(A)f(e), where Xp is the characteristic function of P and f denotes the Fourier transform of f; it has been known for a long time that T is a bounded operator on L"(R2), 1 a > 0, is bounded on L"(R2) for every p between 4/(3 + 2a) and 4/(1 - 2a). The polygon problem (which appears in the work of C. Fefferman as a natural step toward the understanding of the behavior of the Bochner-Riesz multipliers Sa) asks for sharp estimates for the norm of the operator T. Associated to the operator T there is a maximal function M defined by
37 citations
Patent•
26 Oct 1977TL;DR: In a light deflecting device using a polygon mirror, the authors provides a reference surface and a plurality of integral mirrored surfaces formed at a predetermined angle with respect to the reference surface, which is used to produce a flow of air between the reference surfaces and the float members during rotation of the polygon mirrors.
Abstract: In a light deflecting device using a polygon mirror, a polygon mirror member is provided which has a reference surface and a plurality of integral mirrored surfaces formed at a predetermined angle with respect to the reference surface. A fixing member resiliently fixes the polygon mirror member to a drive shaft for rotatively driving the polygon mirror member. The fixing member is effective to vary the angle formed between the reference surface and the drive shaft. A plurality of float members are provided in opposed relationship with the reference surface of the polygon mirror member and fixed independently of the rotation of the polygon mirror member. The float members are effective to produce a flow of air between the reference surface and the float members during rotation of the polygon mirror member to maintain a predetermined distance therebetween.
13 citations
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TL;DR: In an attempt to reveal the breadth of Gauss's interest in geometry, this account is divided into six chapters as mentioned in this paper, where the fundamental theorem of algebra can be proved only with the aid of geometric ideas, and an application of algebra to geometry: the connection between the Fermat primes and the construction of regular polygons.
10 citations
Patent•
21 Jun 1977
TL;DR: In this paper, the size and shape of the gear and its teeth are determined by laying out a regular polygon having an odd number of sides, corresponding to the number of desired gear teeth with the distance from the center of the regular polygnus equal to the pitch radius of the desired gear.
Abstract: This is a compatible gear system and consists of providing an odd number of teeth on each gear, a minimum of five or more. The size and shape of the gear and its teeth are determined by laying out a regular polygon having an odd number of sides, corresponding to the number of desired gear teeth with the distance from the center of the regular polygon equal to the pitch radius of the desired gear. Then a circle is drawn about each polygon corner with a radius equal to one-fourth of the side of the polygon, this series of circles determining the shape and size of the gear teeth. Next, an intermediate series of circles is provided with their centers at the midpoint of the polygon sides and, of necessity, the intermediate circles are tangent to the teeth determining circles. Lines drawn through the points of tangency between each intermediate circle and the adjacent corner circles are necessarily parallel. The outer portions of the corner circles determine the size and shape of the teeth, and the inner portions of the intermediate circles determine the depth of the gear teeth meshing recesses between the teeth. In addition, slight clearances are provided at the tips of the teeth, the clearances being at right angle to a pitch radius extended therethrough. These clearance edges may be a straight chord in the tooth circle, or may be a rounded edge, and, in either case, permit better lubrication. The gears thus made are greater in strength, durability and much more efficient in transmitting power than conventional gears.
9 citations
01 Jul 1977
TL;DR: Extending the Bezier notion of using a polygon to describe a smooth curve, methods of assigning a weight to each vertex which will control the amount of local fit to the polygon of polygonal net are provided.
Abstract: : Of particular importance in an interactive curve and surface design system is the interface to the user. The mathematical model employed in the system must be sufficiently flexible for interaction between designer and machine to converge to a satisfactory result. The mathematical theory of Total Positivity is combined with the interactive techniques of Bezier and Riesenfeld in developing new methods of shape representation which retain the valuable variation-diminishing and convex hull properties of Bernstein and B-spline approximation, while providing improvements in the interactive interface to the user. Specifically, extending the Bezier notion of using a polygon to describe a smooth curve, methods of assigning a weight to each vertex which will control the amount of local fit to the polygon of polygonal net are provided. Thus, the designer can cause cusps and flats easily by manipulating the tension at each vertex. Further, the generalization from curves to surfaces can be done with rectilinear data or triangular data. Illustrations are provided from an experimental implementation of the newly constructed models as a demonstration of their feasibility and utility in computer aided curve and surface design. (Author)
6 citations
Patent•
02 May 1977
TL;DR: In this article, a portable device providing visual images of infra-red sources by means of I.R. detectors/electric signals/light emitting diodes is employed.
Abstract: A portable device providing visual images of infra-red sources by means of I.R. detectors/electric signals/light emitting diodes. Scanning is employed to reduce the numbers of I.R. detector elements and L.E.D.'s, the scanning being effected by a rotating polygon having mirror facets tilted at slightly different angles to scan the object and image fields simultaneously in strips which cover the respective fields in a cycle. One aspect of the invention consists in tilting the I.R. and visual facets oppositely to transfer a low I.R. strip position to a correspondingly high visual strip position and vice versa. A second aspect consists in using a polygon with 45° facets and I.R. detectors and L.E.D.'s on optical paths parallel to the axis of rotation. The scanned strips are then symmetrical about their field center lines.
4 citations
TL;DR: The arrangements of the principal elements of a character generator will he a laser, an acousto-optic deflector and a polygon mirror, and its performance characteristics are calculated and compared to experimental results.
Abstract: The requirements for optical character generation for a commercial electrophotographic high-speed printer—more than 10,000 1pm—are given. They lead to the conclusion that the principal elements of a character generator will he a laser, an acousto-optic deflector and a polygon mirror.The arrangement of a character generator which has been realized in a commercial printer is described in detail. Its performance characteristics are calculated and compared to experimental results.
Patent•
07 Jan 1977
TL;DR: In this paper, the authors used a band of a ceramic fiber mat to heat an oven disc or oven side wall of circumference in a polygon or circle shape by using of a band a ceramic fiber mat.
Abstract: PURPOSE:To heat insulation an oven disc or oven side wall of circumference in a polygon or circle shape by using of a band a ceramic fiber mat.
Patent•
08 Nov 1977
TL;DR: In this article, the installation of multiple ring holders centrically movable at the apices of a polygon facilitates to obtain a system to insert a ring into an annular groove around outside the cylindrical body.
Abstract: PURPOSE:The installation of multiple ring holders centrically movable at the apices of a polygon facilitates to obtain a system to be able to insert a ring into an annular groove around outside the cylindrical body.
Patent•
14 Apr 1977
TL;DR: In this article, a polygon or a circle or an oval or an ellipse is used as the basic profile of the spiral belt buckle instead of the square or rectangular basic profile.
Abstract: The spiral belt buckle is in round material which is wound with approximately a 1 1/4 turn over a rectangular or square cross-section. The winding spacing between the two freely emerging shanks is shaped to draw in a belt. Specifically, instead of the square or rectangular basic profile of the spiral winding, a polygon or a circle or an oval or an ellipse is used as the basic profile. In practical installations, the end of the belt works against a torsion spring.
Patent•
01 Mar 1977
TL;DR: In this article, the upstanding walls have mirror surfaces on their inwardly presented sides so that a viewer, looking through the opened upper end will see the illusion that the upper surface of the floor wall extends infinitely.
Abstract: A device for displaying a sample of sheet material such as carpet or drapery comprises a floor wall having perimetric edges in a polygon shape and a plurality of walls extending upwardly from the perimetric edges thereof. Means are provided for connecting the upstanding walls at the adjoining vertical edges thereof so as to enclose the floor in a box-like configuration having an open upper end. The upstanding walls have mirror surfaces on their inwardly presented sides so that a viewer, looking through the opened upper end will see the illusion that the upper surface of the floor wall extends infinitely.
01 Sep 1977
TL;DR: The feasibility of an optimal real-time algorithm for constructing the convex hull of a set of n points in the plane, using total 0(nlogn) time - which is optimal - with an interpoint delay 0(logn), which gives the real- time property.
Abstract: : In this notebook a collection of new results is presented in computational geometry, which all concern problems of planar geometry. The first problem is that of triangulating a simple n-vertex polygon; we show that this can be done in time 0(nlogn), by first decomposing in time 0(nlogn) the given polygon into a collection of special polygons, called monotone, which can be individually triangulated in time proportional to their numbers of edges. The second result concerns that all-nearest neighbor problem for an n-vertex polygon: a surprising result is that, also no method faster than 0(nlogn) is known for constructing the Voronoi diagram of a convex polygon, the all-nearest-neighbor problem can be solved in time 0(n). Finally, we show the feasibility of an optimal real-time algorithm for constructing the convex hull of a set of n points in the plane. This algorithm constructs the hull by successive updates, using total 0(nlogn) time - which is optimal - with an interpoint delay 0(logn), which gives the real-time property. (Author)
01 Jun 1977
TL;DR: In this article, a method of finding a mapping function which maps the doubly connected region between a circle and a curvilinear polygon onto an annulus is described and numerical results for the ratio of the radii of the annulus given.
Abstract: A method of finding a mapping function which maps the doubly-connected region between a circle and a curvilinear polygon onto an annulus is described and numerical results for the ratio of the radii of the annulus given.
Patent•
14 Mar 1977
TL;DR: In this paper, a polygonal polygon with hinged sides is used to construct a buoyant housing, where supports are connected with fluid sources and producing positive and negative buoyancies in the region of the housing near the supports.
Abstract: The device comprises a housing made in the form of a polygon having hinged sides. Installed in close proximity to housing, supports are tanks connected with fluid sources and producing positive and negative buoyancies in the region of the housing near the supports. To rotate the supports use is made of a means for varying the angles between the housing sides.