Showing papers on "Polygon published in 1985"

Decomposing a Polygon into Simpler Components

Abstract: The problem of decomposing a polygon into simpler components is of interest in fields such as computational geometry, syntactic pattern recognition, and graphics. In this paper we consider decompositions which do not introduce Steiner points. The simpler components we consider are convex polygons, spiral polygons, star-shaped polygons and monotone polygons. We apply a technique for improving the efficiency of dynamic programming algorithms in order to achieve polynomial time algorithms for the problems of decomposing a simple polygon into the minimum number of each of the component types. Using the same technique we are able to exhibit polynomial time algorithms for the problems of decomposing a simple polygon into each of the component types while minimizing the length of the internal edges used to form the decomposition. When the polygons are allowed to contain holes many of the problems become NP-hard.

Storing a collection of polygons using quadtrees

Abstract: An adaptation of the quadtree data structure that represents polygonal maps (i.e., collections of polygons, possibly containing holes) is described ina manner that is also useful for the manipulation of arbitrary collections of straight line segments. The gol is to store these maps without the loss of information that results from digitization, and to obtain a worst-case execution time that is not overly sensitive to the positioning of the map. A regular decomposition variant of the region quadtree is used to organize the vertices and edges of the maps. A number of related data organizations are proposed in an iterative manner until a method is obtained that meets the stated goals. The result is termed a PM (polygonal map) quadtree and is based on a regular decomposition point space quadtree (PR quadtree) that stores additional information about the edges at its terminal nodes. Algorithms are given for inserting and deleting line segments from a PM quadtree. Use of the PM quadtree to perform point location, dynamic line insertion, and map overlay is discussed. The PM quadtree is compared conceptually to the K-structure and the layered dag with respect to typical cartographic data. An empirical comparison of the PM quadtree with other quadtree-based representations for polygonal maps is also provided.

An algorithm to construct continuous area cartograms

Abstract: Continuous area cartograms distort planimetric maps to produce a desired set of areas while preserving the topology of the original map. We present a computer algorithm which achieves the result iteratively with high accuracy. The approach uses a model of forces exerted from each polygon centroid, acting on coordinates in inverse proportion to distance. This algorithm can handle more realistic descriptions of polygon boundaries than previous algorithms and manual methods, thus enhancing visual recognition.

Visibility-polygon search and euclidean shortest paths

Abstract: Consider a collection of disjoint polygons in the plane containing a total of n edges. We show how to build, in O(n2) time and space, a data structure from which in O(n) time we can compute the visibility polygon of a given point with respect to the polygon collection. As an application of this structure, the visibility graph of the given polygons can be constructed in O(n2) time and space. This implies that the shortest path that connects two points in the plane and avoids the polygons in our collection can be computed in O(n2) time, improving earlier O(n2 log n) results.

Frequency Polygons: Theory and Application

Abstract: In this article I investigate the theoretical properties and applications of the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. For estimating an unknown probability density function using a random sample, the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error, achieving the same rate of convergence to zero of the integrated mean squared error as non-negative kernel estimators. Data-based algorithms for constructing frequency polygons are discussed and illustrated. One is based on a histogram with bin width equal to 2.15sn -1/5, where s is an estimate of the standard deviation from a sample of size n. Another approach is based on the method of generalized crossvalidation. The bivariate frequency polygon is also investigated. Comparisons are made between the frequency polygon and other density estimators. Examples with data in one and two dimensions are presented.

Shading apparatus for displaying three dimensional objects

Abstract: An apparatus for shading a polyhedron at high speed is disclosed which includes the combination of a polygon-scan line conversion processor and an inner product interpolation processor for obtaining a pair of inner products of vectors indicative of a relation among the direction of a normal, the direction of a light source and the direction of a view point each viewed at a point within a polygon having a plurality of vertices, on the basis of the position of said point in the polygon and the direction of a normal at each of the vertices, a table searched on the basis of the inner products of vectors and holding a series of brightness data which have previously been calculated for a series of values of each of the inner products of vectors, a buffer for storing the result of table search for the above table, and a D/A conversion circuit for converting the result of table search into a signal which is used as a brightness control signal in a display device.

Curvilinear dynamic image generation system

Abstract: A system is disclosed for the production of real-time, computer-generated images as for projection on a curvilinear surface as on the inside of a dome. A general-purpose computer provides data for visual images and supplies the data in an ordered format. Such data is processed first with regard to viewpoint and second with regard to channel operation. Specifically, viewpoint processing incorporates an object manager and a polygon manager. Channel processing includes a geometric processor and a display processor. The display processor drives a projector for illuminating a spherical surface with the desired images. The geometric processor incorporates several well known elements including a rotator, clippers, a slope calculator and polygon buffer. Additionally, the geometric processor incorporates a segmenter and mapper which replaces the traditional perspective divider to process the data so as to generate images for curvilinear projection. The segmenter tests the character of polygon edges in spherical coordinate format and corrections are performed on data in the same format. Segmentation is performed on data in a rectangular coordinate format.

A fast algorithm for polygon containment by translation

Abstract: The polygon containment problem is the problem of deciding whether one polygon, C, can be translated to fit within another polygon N. We present an algorithm that runs in time O (cn log cn) to solve this problem, in the case that the polygon C is convex. Here c is the number of bounding edges of C, and n is the number of bounding edges of N. The algorithm actually computes the feasible region, that is, a description of the set of all placements of C inside N. The algorithm is close to optimal in that the feasible region may have O (cn) vertices.

High resolution laser printer

Abstract: A high resolution laser printer utilizes a first cylindrical lens to focus a laser beam in the sagittal direction prior to the facets of a rotating polygon. The light reflected from the polygon is focused in the horizontal direction at the photoconductor drum of an electrophotographic process by a scan lens assembly. Following the scan lens assembly, there is a second cylindrical lens which also focuses the laser beam in the horizontal direction at the photoconductor. The first and second cylindrical lenses are positioned to eliminate errors in the facets of the polygon. Scan velocity deviations resulting from the optical design are corrected by storing correction information in a memory and using this information to vary the rate at which the laser is modulated with data.

Self-avoiding polygons on the square, L and Manhattan lattices

Abstract: Transfer-matrix techniques are used to extend the self-avoiding polygon generating function on the square lattice to terms in x46, corresponding to 46 step polygons. These techniques are then extended to apply to directed square lattices, such as the L and Manhattan lattice, and the self-avoiding polygon generating function to x48 is found for these lattices. Series analysis confirms that the 'specific heat' exponent alpha =1/2 for the self-avoiding walk problem, and gives the following estimates for the connective constants: mu (SQ)=2.638155+or-0.000025, mu (L)=1.5657+or-0.0019 and mu (Man.)=1.7328+or-0.0005. Some evidence for a correction to scaling exponent Delta approximately=0.84 is found from square lattice series.

Fast triangulation of the plane with respect to simple polygons

Abstract: Let P 1 ,…, P k be pairwise non-intersecting simple polygons with a total of n vertices and s start vertices A start vertex, in general, is a vertex both of which neighbors have larger x coordinate We present an algorithm for triangulating P 1 ,…, P k in time O ( n + s log s ) s may be viewed as a measure of non-convexity In particular, s is always bounded by the number of concave angles + 1, and is usually much smaller We also describe two new applications of triangulation Given a triangulation of the plane with respect to a set of k pairwise non-intersecting simple polygons, then the intersection of this set with a convex polygon Q can be computed in time linear with respect to the combined number of vertices of the k + 1 polygons Such a result had only be known for two convex polygons The other application improves the bound on the number of convex parts into which a polygon can be decomposed

Finding minimal convex nested polygons

Abstract: We consider the problem of finding a polygon nested between two given convex polygons that has a minimal number of vertices. Our main result is an O(nlogk) algorithm for solving the problem, where n is the total number of vertices of the given polygons, and k is the number of vertices of a minimal nested polygon. We also present an O(n) sub-optimal algorithm, and a simple O(nk) optimal algorithm.

Scanning systems with polygon scanner having curved facets

Abstract: A raster scanner system having double reflection from polygon facet surfaces which have aspheric curvature In addition to providing a scan which is free of astigmatism and bow, the aspheric surfaces provide the scan and focus functions of conventional scan lens optics

High speed pattern generator for electron beam lithography

Abstract: A pattern generator for supplying beam deflection and blanking signals in an electron beam lithography system which writes polygon pattern features by sweeping a beam of rectangular cross-section over each polygon and simultaneously varying the length of the rectangular cross-section. The pattern generator converts polygon size and shape data to an upper shape signal and a lower shape signal. The shaping signals are subtracted to provide a beam length signal. The lower shape signal controls the beam position during writing of the polygon. The pattern generator further includes a ramp generator for sweeping the beam over the polygon. The ramp signal and shaping signals are synchronized by detecting the points in the sweep at which polygon turn points occur. The shape signal generators utilize interleaved operation for high speed. A blanking circuit provides uniform exposure of pattern features by controlling the width of the rectangular beam. The beam is ramped on and off at a rate which matches the rate of the sweep signal.

Process for graphically representing a structure

Abstract: A process for graphically representing a simulated workpiece as processed by a numerically-controlled machine tool Stored data defining the workpiece and the tool as well as program data defining relative movement therebetween are used to generate polygons which represent the workpiece and the tool and the path of movement of the tool These polygons are hierarchically ordered and liked Corresponding to the hierarchic order of the polygons, intersection point calculations in the linkage are made only for the workpiece polygons which overlap the tool polygon The image of the processed workpiece is generated successively through the continuous superposition of the changing workpiece polygons with the processing polygons corresponding to the tool and its path Workpiece edges which drop out are erased, and invisible workpiece edges are nto represented

A Linear Time Algorithm for Triangulating a Point-Visible Polygon

Abstract: The triangulation of a point-visible (star-shaped) polygon cannot be performed trivially if its kernel does not share a vertex with the polygon. The paper presents a triangulation algorithm that exploits point-and strong edge-visibility. It is through these two properties that the authors are able to triangulate in linear time. After classifying simple polygons by visibility, the authors show that strongly edge-visible polygons can be triangulated in linear time. A point-visible polygon is transformed into a strongly edge-visible polygon by the following steps: partitioning with a ray, partially triangulating both partitions, merging the two remaining polygons, and showing that the merged polygon is a strongly edge-visible polygon

Linear Quadtrees from Vector Representations of Polygons

Abstract: A new algorithm is presented which produces various forms of linear quadtrees directly from a vector representation of a polygon. This algorithm takes advantage of specific properties of linear quadtrees and associated linear keys to infer the colors of all parts of the region not cut by the polygon boundary. The method is further extended to multicolored (rather than binary) linear quadtrees which may be useful in geographic information systems applications.

Polygon fill processor

Abstract: Method and apparatus for filling convex polygons for display in a raster graphics system is described. Starting from the bottom of the polygon, horizontal lines are drawn to fill the polygon as y-value increments. A polygon-fill processor includes two modified Breshenham line generators.

A Fast Algorithm for Polygon Containment by Translation (Extended Abstract)

Abstract: The polygon containment problem is the problem of deciding whether one polygon, C, can be translated to fit within another polygon N. We present an algorithm that runs in time O (cn log cn) to solve this problem, in the case that the polygon C is convex. Here c is the number of bounding edges of C, and n is the number of bounding edges of N. The algorithm actually computes the feasible region, that is, a description of the set of all placements of C inside N. The algorithm is close to optimal in that the feasible region may have O (cn) vertices.

Laser printer having means for changing the output-image size

Abstract: A laser printer is disclosed which uses a rotating polygon and a cylindrical mirror disposed between the polygon and an image zone. The cylindrical mirror corrects for pyramidal errors in an output image formed on a moving photosensitive member disposed at the image zone by making the operative polygon facet and the image zone optically conjugate in the page scan direction. To change the length of the image in the line scan direction at the image zone, the distance between the polygon and image zone is varied. This is accomplished by moving the cylindrical mirror or by positioning a different cylindrical mirror at a new optical path position. The velocity of the photosensitive member is adjusted to change the length of the output image in the page scan direction. The tilt angle of the cylindrical mirror moved to the new optical path position is changed or the cylindrical mirror is removed from the path and a different cylindrical mirror is inserted at a new optical path position. In either case, at the new image format size, the cylindrical mirror makes the operative polygon facet optically conjugate with the image zone in the page scan direction to correct for pyramidal errors. Anamorphic beam shaping optics are provided which preceed the polygon to form and position the line scan waist so that it is adjacent to the image zone at different line scan lengths, and to change the beam size at the polygon in the page scan direction at different page scan lengths.

A subjective judgment study of polygon based curved surface imagery

Abstract: In the past computer graphics efforts, several researchers have demonstrated that polygon models can be used to produce images of curved surfaces that appear to be smooth and accurate. However, the authors know of no attempt to appraise such imagery by using multiple human observation ratings.The effectiveness of curved surface imagery generated from polygon models was investigated in a judgment study. Research subjects evaluated sphere model imagery derived from several polygon densities and shading procedures including flat shading, shade interpolation (Gouraud) and normal interpolation (Phong). Results of the evaluations indicated that little was gained by reducing the average polygon areas below approximately 110 pixels per polygon for spheres of 95 pixel radii displayed on a 512 x 512 resolution monitor. Evaluations for both shade and normal interpolution placed polygon image quality reasonably close to an “ideal” image. Although the evaluations indicated that normal interpolation was slightly superior to the shade interpolation, shade interpolation required significantly less computation. Most significantly, results from this study provide strong support for the notion that polygons can be used effectively to produce smooth shaded imagery of curved surface models.

Method of graphing cardiopulmonary data

Abstract: A method of graphically displaying cardiopulmonary function data by providing a plurality of axes radially projecting from a common point of origin, there being one such axis for each cardiopulmonary parameter to be plotted, each axis having scale markings thereon whereby the normal value for each parameter will lie an equal distance from said point of origin, and straight line segments connected between said normal values on adjacent axes will form a generally regular polygon. The cardiopulmonary function data to be displayed are then plotted on the same appropriate axis and to the same scale such that deviations of any such data from normal will result in an irregular polygon when the segments are joined between plotted values on adjacent axis. Deviations of the irregular polygon from the regular polygon thus provides a clinician with a readily visible display of possible cardiopulmonary disfunction.

Interpolating Patches Between Cubic Boundaries

Abstract: We describe the theoretical frame for a method of creating and describing rounded objects of arbitrary topology in CAD, and its implementation for UNIGRAFIX, a polygon-based modeler developed at UC Berkeley that generates black-and-white, smooth-shaded images on several output devices. The mathematical foundation for building triangular patches interpolating cubic edges and blending with geometric continuity is given, and various approaches are discussed. To represent curvature information, we extended the UNIGRAFIX language to UniCubix, and we implemented uci, an interactive shell that interprets a UniCubix description and converts it into UNIGRAFIX wireframes or polyhedral nets that approximate curved patches. Uci also provides a prototype of a global smoothing operation, that takes a polyhedral object of arbitrary topology and creates the UniCubix representation of a smooth object interpolating the input vertices.

A vision system to identify occluded industrial parts

Abstract: A vision system is presented that recognizes occluded industrial parts. The unknown image may contain multiple objects that may touch or overlap giving rise to partial occlusion. The vision system uses stored models to locate and identify the objects in the scene. The models are based on the boundary of the object, since we assume that the objects are rigid and planar. From the polygon approximation of the boundary, vertices of high curvature are identified as "corners." These corners are used as features in detecting the model in the image. A globally consistent coordinate transform that takes the model into the image is found by using a Hough like transform and the corner features.

Display of Visible Edges of a Set of Convex Polygons

Abstract: We present an efficient algorithm for computing visible boundary edges of a set of m nonoverlapping convex polygons The algorithm runs in time O(m log m + Σ m i=1 log n i + F), where n i is the number of vertices of polygon i and F the complexity of the visible scene The method can be applied to solve three dimensional hidden-surface problem based on scan-line approach, namely, each scan-line plane intersects a set of objects in three-dimensional space and yields a set of nonoverlapping convex polygons

Rotational polygon mirror and method of manufacturing the same

Abstract: A rotational polygon mirror and a method of manufacture thereof comprising a base member having a polygonal pole-like portion, the base member being formed of a relatively light-weight material which can be easily molded into a complicated shape, and a coating film which is provided on one or more side surfaces of the polygonal pole-like portion and on which a reflecting mirror surface is formed, the coating film being of at least one layer structure.