Showing papers on "Polygon published in 1978"

A hidden-surface algorithm with anti-aliasing

Abstract: In recent years we have gained understanding about aliasing in computer generated pictures and about methods for reducing the symptoms of aliasing. The chief symptoms are staircasing along edges and objects that pop on and off in time. The method for reducing these symptoms is to filter the image before sampling at the display resolution. One filter that is easy to understand and that works quite effectively is equivalent to integrating the visible intensities over the area that the pixel covers. There have been several implementations of this method - mostly unpublished - however most algorithms break down when the data for the pixel is complicated. Unfortunately, as the quality of displays and the complexity of pictures increase, the small errors that can occur in a single pixel become quite noticeable. A correct solution for this filter requires a hidden-surface algorithm at each pixel! If the data at the pixel is presented as a depth-ordered list of polygons then the average visible intensity can be found using a polygon clipper in a way similar to that employed by two known hidden-surface algorithms. All of the polygons in a pixel are clipped against some front unclipped edge into two lists of polygons. The algorithm is recursively entered with each new list and halts when the front polygon is clipped on all sides, thereby obscuring the polygons behind. The area weighted colors are then returned as the value to be added to the other pieces in the pixel.

Polygon shadow generation

Abstract: A general purpose method for generating shadows using a polygonal coordinate data base is presented. The method is based on an object space polygon clipping hidden surface removal algorithm. Output from the program is in the same three-dimensional polygon format as the input. Thus, a shadowed data environment may be easily created and viewed from any observer position with no additional depth sorting time required for the hidden surface removal process. Shadows can also be cast by more than one light source. Since the shadows are generated in object space, the results can be used for both visual display and numerical analysis.

Decomposition of Polygons into Convex Sets

Abstract: A method is presented for decomposing polygons into convex sets. The method is based upon a Delaunay tessellation of the polygon. It is implemented as a divide-and-conquer technique.

Billiards in Polygons

Abstract: Some questions concerning the orbits of a billiard ball in a polygon are studied. It is shown that almost all such orbits come arbitrarily close to a vertex of the polygon, implying that the entropy of the corresponding geodesic flow is zero. For polygons with rational angles, we show by using interval exchange transformations that almost all orbits are spatially dense. Two applications are given.

Clipping polygon faces through a polyhedron of vision

Abstract: A flight simulator combines flight data and polygon face terrain data to provide a CRT display at each window of the simulated aircraft. The data base specifies the relative position of each vertex of each polygon face therein. Only those terrain faces currently appearing within the pyramid of vision defined by the pilots eye and the edges of the pilots window need be displayed at any given time. As the orientation of the pyramid of vision changes in response to flight data, the displayed faces are correspondingly displaced, eventually moving out of the pyramid of vision. Faces which are currently not visible (outside the pyramid of vision) are clipped from the data flow. In addition, faces which are only partially outside of pyramid of vision are reconstructed to eliminate the outside portion. Window coordinates are generated defining the distance between each vertex and each of the boundary planes forming the pyramid of vision. The sign bit of each window coordinate indicates whether the vertex is on the pyramid of vision side of the associated boundary panel (positive), or on the other side thereof (negative). The set of sign bits accompanying each vertex constitute the "outcode" of that vertex. The outcodes (O.C.) are systematically processed and examined to determine which faces are completely inside the pyramid of vision (Case A--all signs positive), which faces are completely outside (Case C--All signs negative) and which faces must be reconstructed (Case B--both positive and negative signs).

Variable resolution for real-time simulation of a polygon face object system

Abstract: The visual system within an aircraft flight simulator receives flight data from a flight simulation computer, and polygon face terrain data from a gaming area data base. The visual system computer organizes this image data into the prescribed format, and forwards the image data to an image processor. The following types of data are included in the format: Position vectors (Vp), defining the changing position of the aircraft with respect to the origin of the terrain coordinate system. Rotational data (H matrix), defining the changing attitude of the aircraft with respect to the terrain coordinate system. Cif data, defining color, intensity, defocus, etc. of individual faces. Variable resolution data (Var Res) defining the scan line density (vertical display resolution) for each polygon face is included in the CIF data. The vertices of each face pass through the image processor as a unit. Each face is displayed on a CRT by a separate miniraster which is caligraphically positioned and which sweepes the electron beam to paint the polygon interior defined by the face vertices. Each face has a uniform color and intensity as determined by the CIF data. Further, each face has a particular vertical resolution determined by the detail and importance of the face, and the time available to present the scan lines.

Computational Morphology: Three-Dimensional Computer Graphics for Electron Microscopy

Abstract: This paper describes a method for the computer reconstruction of surfaces from a sequence of electron micrographs, and a data structuring approach to the problem of representing and analyzing objects of physiological importance. The reconstruction technique involves the following stages: 1) object outlines are traced from each section, 2) the computer chain encodes these outlines, 3) the chain codes are reduced to the minimum number of boundary points which satisfactorily define the boundary, 4) polygons are mapped onto the boundary points between sections to approximate the surface, and 4) color coded, shaded surface views are computed of any subset of objects viewed and illuminated from arbitrary locations.

Apparatus and method for correcting imperfection in a polygon used for laser scanning

Abstract: An electro-optical light scanning system using a modulated laser illumination source directed upon a multifaceted rotating polygonal mirror or polygon. The mirrored facets reflect the impinging light toward a moving photoreceptor and forms a raster of scan lines as the photoreceptor moves. The system incorporates sensing optics and closed loop electronics for correcting inaccuracies in the position of the reflected light resulting from defects in the angular relationship between the plane of the facets and that of the rotating axis of the polygon as well as those errors due to inherent angular misalignment between each of the facets of the rotating polygon. The invention also encompasses amplitude modulation for varying the intensity of the laser illumination in conjunction with the acousto-optical modulation for maintaining a constant level illumination and/or for varying the spot size.

Polygon shadow generation

Abstract: A general purpose method for generating shadows using a polygonal coordinate data base is presented. The method is based on an object space polygon clipping hidden surface removal algorithm. Output...

A scan line algorithm for computer display of curved surfaces

Abstract: The conventional procedure for generating shaded images of curved surfaces is to approximate each surface element by a mosaic of polygons and to then apply one of several established polygon display algorithms. The method described here extends these polygon based techniques to produce an excellent approximation of bi-cubic parametric surfaces in scan line order.Each surface patch is described in terms of cubic edge curves, including parametric curves on the interior of the patch as well as the patch boundaries. Specifying interior edges has the effect of subdividing the patch and generally results in a more accurate image. The silhouette, approximated by a piecewise cubic interpolant, further divides the patch into front facing and rear facing portions.The edge curves are intersected by successive scanning planes to form the endpoints of scan line segments. Depth and surface normal are linearly interpolated between the endpoint values. Visibility is calculated for each segment by a hybrid priority/z-buffer scheme. Shading is computed using Phong's illumination model with the interpolated surface normal.The algorithm is currently used for display of B-spline surfaces as part of an experimental display processor.

Stability of body‐centered polygonal configurations of ideal vortices

Abstract: Thomson’s results concerning the stability of regular vortex polygons are generalized to include the case of the polygonal center being occupied by a vortex of arbitrary relative strength. Of most significance is the fact that a strong enough central vortex can stabilize a Thomson polygon consisting of arbitrarily many vortices. The efficacy of the free energy criterion for stability is demonstrated in showing that the Thomson heptagon is stable.

Simulation of an object system formed by polygon faces having a series of fundamental shapes and dimension

Abstract: The terrain data within an aircraft flight simulation system includes two and three dimensional object items formed by polygon faces having shapes orientations and dimensions selected from a series of fundamental shapes orientations and dimensions. The image data is coordinated in a buffer memory according to a particular format, and forwarded to an image processor. Instruction words appear throughout the format for identifying the type of data in the data words following each instruction word, and for controlling the flow of data through the image processor. The image data formatted and processed includes: Initial position data (Vo), defining the position of initial points or landmark points (i.e., the first vertex of each polygon face) within the terrain coordinate system. Shape data (delta data) defining the relative position of other points, (i.e., the remaining vertices of the fundamental shape incorporated in each polygon face) with respect to the landmark points of each Vo. Scale data for defining the dimensions of each face to be formed from the fundamental object shapes. The image processor translates initial position data into the aircraft coordinate system; then rotates, clips, and projects each three-dimensional position vector to form two-dimensional display vectors. Raster scanlines are generated from the display vectors Vd of each face for display on a CRT.

The Differential Effects of the Use of Manipulative Aids on the Learning of Geometric Concepts by Elementary School Children

Abstract: There is considerable support for the use of manipulative aids when teaching geometric concepts (Cheatham, 1969; Dienes & Golding, 1967; Yakimanskaya, 1971). However, the question of what aids to use and how to use them most effectively is unanswered. Wiviott's (1970) study suggests that concepts be presented to younger students in terms of the bases of classification they are able to use, that is, square, rectangle, rhombus, parallelogram, quadrilateral, triangle, circle, and cube. This research attempted to determine which instructional setting is most effective in teaching geometry to students at the third-grade level. The concepts (selected after examining the research by Dowell [1963] and Peck [1970]) included in this research were the following: point, line, line segment, triangle, square, pentagon, ray, angle, congruence of lines, congruence of angles, polygon, cube, tetrahedron, face, vertex, edge, diagonal, side, and endpoint.

3-D graphic display of discrete spatial data by prism maps

Abstract: An efficient algorithm for displaying 3-D scenes showing a discrete spatially varying surface is described. Given a 2-D map or planar graph composed of polygons where each polygon has a positive real number attribute, a prism is erected on each polygon with height proportional to that attribute. The resulting 3-D scene is plotted with shading and hidden lines removed. Thus the spatial variation of the attribute may be quickly and intuitively grasped by the nontechnical observer. This has applications to areas such as geography if the map is a cartographic map, or to physics if the map diagrams the periodic table. The algorithm takes time O(N*log(N)) where N is the number of edges in the map. Most of the calculations can be done without knowing the prism heights so extra plots with different attributes for the prisms can be produced quickly. This algorithm has been implemented and tested on maps of up to 12000 edges.

Repeating pattern simulation of a polygon face object system

Abstract: The terrain objects within the data base for an aircraft flight simulation system are formed by polygon faces and include certain objects formed by repetitive patterns of equally spaced identical polygon faces. More specifically, objects such as runway markings, which are a series of elongated narrow rectangles equally spaced along the length of the runway. The image data is arranged in a particular format including instruction words for identifying the type of data in the data words following each instruction word, and for controlling the flow of data through the image processor. The image data formatted and processed includes: Initialization vectors (Vo), defining the position of landmark points (i.e., the first vertex of the initial polygon of each series of faces). Delta data (shape data), defining the position of other points, (i.e., the remaining vertices of each initial face) with respect to the initial point therefor. Increment data defining the uniform spacing between adjacent identical faces of each series. Series data defining the number of identical faces in each series. The image processor translates initial position data into the aircraft coordinate system, then rotates, clips, and projects each three-dimensional position vector to form two-dimensional display vectors (Vd). Raster scanlines are generated from the display vectors Vd of each face for display on a CRT.

A constrained isoperimetric problem

Abstract: Of all sets lying within a given convex plane set S, which one gives the greatest ratio of area A to perimeter P? This was posed to one of us by E. Bombieri for the case when S is a square. He asserted that T. Bang used an easy upper bound on A/P in an elementary proof of the Prime Number Theorem. After solving this problem from scratch, we learned from M. Silver that Besicovitch (1) had solved a closely related problem, which we call Besicovitch's problem and which will be discussed shortly. Later, we found that the problem of maximizing A/P had occurred to Garvin (4, 5) and that he solved the problem when S is a triangle. Garvin inspired DeMar (2) to consider and solve Besicovitch's problem for the triangle and for polygons. DeMar found that Steiner (8, p. 166) had solved Besicovitch's problem for triangles, but Steiner was only interested in subsets which touched all sides of a polygon (8, p. 168). More recently, J. Wills has referred us to Herz and Kaapke (6) who use Besicovitch's results to solve our problem of maximizing A/P for polygons circumscribed about a circle.

Computation of the Histograms of the Number of Edges and Neighbours of Cells in a Tessellation

Abstract: The spatial random structure studied is a population of curvilinear polygons partitioning the space. The object of the paper is to provide an automatic method to compute the histograms of the number of edges and neighbours per polygon. The method consists of three steps: first, polygon boundaries are skeletonized; then, a procedure to compute the number of edges and polygon neighbours is outlined. Finally, we develop a statistical method given by MILES to compensate edge effects in planar sampling. Practical applications are made on cells of a sample of sintered alumina.

Tension cord burglar-alarm

Abstract: Members are attachable to the window or door around which a cord can be stretched, under tension, in the form of a polygon. A switch is positioned within the polygon, said switch being attached by further cords to two of the sides of the polygon. The switch is held under a selected tension and the shape of the polygon is such that an intruder cannot enter through the window or door without disturbing the cord polygon and, thereby, the tension on the switch. The switch is designed to respond to a change in the tension applied thereto, to supply a signal to an alarm system.

On an isoperimetric problem

Abstract: Keywords: polygon inscribed into a circle ; greatest area ; convex hull Note: Professor Pach's number: [005] Reference DCG-ARTICLE-1978-001 Record created on 2008-11-14, modified on 2017-05-12

A scan line algorithm for computer display of curved surfaces

Abstract: The conventional procedure for generating shaded images of curved surfaces is to approximate each surface element by a mosaic of polygons and to then apply one of several established polygon displa...

Cutter holding device in a polygon cutting apparatus

Abstract: A rotary cutter holding device for use with a polygon cutting apparatus in a movable spindle-type automatic lathe is provided The cutter holding device utilizes a pair of angular contact ball bearings in conjunction with a belleville spring to carry the thrust load imposed by the cutter

The Polygon Package

Abstract: The memoranda listed below describe the polygon package in its present state. For easy reference, these memoranda have been packaged together. Some of the material included is obsolete. In general, the later-dated material is better than the earlier material. Some specific hazards are indi~ated in the index below.

A computer-assisted graphical method for identification and correlation of igneous rock chemistries

Abstract: We have devised a computer-assisted graphical method for correlating chemical analyses in suites of related igneous rocks. The method provides a direct and empirical means of sample identification using all of the reported chemistry. In a study of basalt of the Columbia River plateau, the method has been used for (1) checking field identifications of rocks, (2) spotting analytical errors in analyses of samples of known chemical type, (3) identifying repetitions of chemistry in thick stratigraphic sections, and (4) matching analyses of dikes to those of flows that the dikes may have fed. Program OXVAR creates a set of oxide-oxide variation diagrams on a CALCOMP plotter; any oxide may be used as the base (for example, MgO, SiO2). For each oxide-oxide variation diagram, program CTSET creates the smallest convex polygon that encloses all the data points as well as a larger polygon that takes into account the error in determination of each data point. A set of nested polygons, one for each oxide, then represents a chemical type. Program CTYPE identifies unknown analyses by comparing them with all defined chemical types. If all the elements or oxides of an analysis plot within the defined polygons for a given chemical type, then an identification is printed. If at least one element of an analysis falls outside the set of polygons defined for each chemical type, then “unidentified” is printed, and qualitative information on the amount of deviation from the defined chemical types is given.

Recorder by means of laser beam

Abstract: PURPOSE:To obtain the video recording with high quality and a good linearity, by eliminating non-linear distortion of of the picture developed with the use of rotary polygon

The interactive digitizing of polygons and the processing of polygons in a relational database

Abstract: A system is described for the interactive digitizing of polygons for such two-dimensional spatial modeling applications as growth studies and thematic map production. A digitizer connected online to a mini-computer with a refresh display is used for capturing polygon data from source documents. An area boundary network is developed by digitizing nodes, pointing to node pairs, and then digitizing edges only once. A CRT display monitors, and bell-rings confirm input operations. Facilities to erase nodes or edges are available. From the nodes and edges, software produces polygons which the user may name or discard. Slivering and overlap are eliminated because contiguous polygons have identical edge descriptions. Named polygons are then transferred to a large university time-sharing computer where spatial editing operations may be carried out, such as unioning pieces of a large map. The named polygons may he given additional non-spatial attributes and entered into a relational type of data base.