Showing papers on "Polygon published in 1986"

Computational complexity of art gallery problems

Abstract: We study the computational complexity of the art gallery problem originally posed by Klee, and its variations. Specifically, the problem of determining the minimum number of vertex guards that can see an n -wall simply connected art gallery is shown to be NP-hard. The proof can be modified to show that the problems of determining the minimum number of edge guards and the minimum number of point guards in a simply connected polygonal region are also NP-hard. As a byproduct, the problem of decomposing a simple polygon into a minimum number of star-shaped polygons such that their union is the original polygon is also shown to be NP-hard.

Visibility of disjoint polygons

Abstract: Consider a collection of disjoint polygons in the plane containing a total ofn edges We show how to build, inO(n 2) time and space, a data structure from which inO(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 inO(n 2) 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 inO(n 2) time, improving earlierO(n 2 logn) results

Image rendering by adaptive refinement

Abstract: This paper describes techniques for improving the performance of image rendering on personal workstations by using CPU cycles going idle while the user is examining a static image on the screen. In that spirit, we believe that a renderer's work is never done. Our goal is to convey the most information to the user as early as possible, with image quality constantly improving with time. We do this by first generating a crude image rapidly and then adaptively refining it where necessary as long as the user does not change viewing parameters. The renderer operates in a succession of phases, first displaying only vertices of polygons, next polygon edges, then flat shading polygons, then shadowing polygons, then Gouraud shading polygons, then Phong shading polygons, and finally anti-aliasing. Performance is enhanced by each phase using results from previous phases and trimming the amount of data needed by the next phase. In this way, only a fraction of the pixels in an image may be Phong shaded while the rest may be Gouraud or flat shaded. Similarly anti-aliasing is performed only on pixels around which there is significant color change. The system features fast response to user intervention, encourages user intervention at any moment, and makes useful the idle cycles in a personal computer.

Optimum watchman routes

Abstract: In this paper we consider the problem of finding shortest routes from which every point in a given space is visible (watchman routes). We show that the problem is NP-hard when the space is a polygon with holes even if the polygon and the holes are convex or rectilinear. The problem remains NP-hard for simple polyhedra. We present O(n) and O(nlogn) algorithms to find a shortest route in a simple rectilinear monotone polygon and a simple rectilinear polygon respectively, where n is the number of vertices in the polygon. Finding optimum watchman routes in simple polygons is closely related to the problem of finding shortest routes that visit a set of convex polygons in the plane in the presence of obstacles. We show that finding a shortest route that visits a set of convex polygons is NP-hard even when there are no obstacles. We present an O(logn) algorithm to find the shortest route that visits a point and two convex polygons, where n is the total number of vertices.

A General Version of Crow's Shadow Volumes

Abstract: In 1977 Frank Crow introduced a new class of algorithm for the generation of shadows. His technique, based on the concept of shadow volumes, assumes a polygonal database and a constrained environment. For example, polyhedrons must be closed, and polygons must be planar. This article presents a new version of Crow's algorithm, developed at the Universite de Montreal, which attempts a less constrained environment. The method has allowed the handling of both open and closed models and nonplanar polygons with the viewpoint anywhere, including any shadow volume. It does not, however, sacrifice the essential features of Crow's original version: penetration between polygons is allowed and any number of light sources can be defined anywhere in 3D space, including the view volume and any shadow volume. The method has been used successfully in the film Tony de Peltrie and is easily incorporated into an existing scan-line, hidden-surface algorithm.

A fast shaded-polygon renderer

Abstract: Image rendering is the performance bottleneck in many computer-graphics systems today because of its computation-intensive nature. Described here is a one-chip VLSI implementation of a shaded-polygon renderer which provides an affordable solution to the bottleneck. The chip takes advantage of a unique extension to Bresenham's vector drawing algorithm [1] to interpolate four axes (for Red, Green, Blue and Z) across a polygon, in addition to the X and Y values. Its inherent accuracy and ease of high-speed hardware implementation distinguish this new algorithm from interpolation with incrementing fractions (DDA).This chip was designed as part of a workstation primarily for mechanical engineering CAD applications. The pipelining and internal bandwidth possible on the chip allows rendering speeds of over twelve-thousand, 1000-pixel, shaded polygons per second, suitable for interactive manipulation of solids. Described in this paper is the derivation of the new algorithm and its implementation in a pipelined, polygon-rendering chip.

Minimum polygonal separation

Abstract: In this paper we study the problem of polygonal separation in the plane, ie, finding a convex polygon with minimum number k of sides separating two given finite point sets (k-separator), if it exists We show that for k = Θ(n), Ω(n log n) is a lower bound to the running time of any algorithm for this problem, and exhibit two algorithms of distinctly different flavors The first relies on an O(n log n)-time preprocessing task, which constructs the convex hull of the internal set and a nested star-shaped polygon determined by the external set; the k-separator is contained in the annulus between the boundaries of these two polygons and is constructed in additional linear time The second algorithm adapts the prune-and-search approach, and constructs, in each iteration, one side of the separator; its running time is O(kn), but the separator may have one more side than the minimum

Geometric retrieval problems

Abstract: A large class of geometric retrieval problems has the following form. Given a set X of geometric objects, preprocess to obtain a data structure D(X). Now use D(X) to rapidly answer queries on X. We say an algorithm for such a problem has (worst-case) space-time complexity O(f(n), g(n)) if the space requirement for D(X) is O(f) and the “locate run-time” required for each retrieval is O(g). We show three techniques which can consistently be exploited in solving such problems. For instance, using our techniques, we obtain an O(n2 + e/log n, log n log(1/e)) space-time algorithm for the polygon retrieval problem, for arbitrarily small e, improving on the previous solution having complexity O(n7, log n).

Method of robust stability analysis with highly structured uncertainties

Abstract: This paper presents a method of analyzing the stability of linear time-invariant interconnected systems with uncertainties: each subsystem is single-input single-output and its vector locus lies inside a polygon at each frequency. It is shown that the convex closure of the image of the Cartesian product of these polygons under the mapping \phi = \det [I + FH] agrees with the convex closure of the image of the vertices of these polygons under the mapping φ. From this result the image can be estimated by a finite Dumber of calculations. A sufficient stability condition is obtained by applying this result to the multivariable Nyquist stability criterion.

ON THE p-ADIC THEORY OF EXPONENTIAL SUMS

Abstract: where V is an arbitrary affine variety defined over Fq, V(Fq) denotes the Fq-rational points of V, f is a regular function on V, and Tq is an arbitrary additive character on Fq. The estimate (1.2) is given in terms of the dimension of the ambient space, and the degree of polynomials which definef and V. The work generalizes the result of Katz [4] which gives a best-possible estimate for the sum(*) in the case Tq is the trivial character (equivalently, Tq is non-trivial but f is the zero function). Thus Katz's estimate is a best-possible estimate for the p-divisibility of N(V) the number of Fq rational points of V. These questions on p-divisibility trace their origins to a problem posed by Artin and solved by Chevalley and Warning [10]. The results may be interpreted as specifying a sharp lower bound for the "first slope" of the Newton polygon of the L-function associated with the sum(*). A finer measure of the p-adic behavior of this L-function is given by the shape of its Newton-polygon. Dwork, in [3], showed that for nonsingular projective hypersurfaces the Newton polygon of the zeta function lies over its Hodge polygon. Katz's result in [4] showed that the same relationship holds for the first slopes of the two polygons in the case of a nonsingular projective complete intersection; at the same time, he formulated a general conjecture which was subsequently proved by Mazur [6] using crystalline cohomology. In section 2, we prove the analogue for exponen-

An algorithm for covering polygons with rectangles

Abstract: Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP-hard. However, we give here an O(v2) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Gyori (J. Combin Theory Ser. B 37, No. 1, 1–9).

Minimally covering a horizontally convex orthogonal polygon

Abstract: In this paper we present O(n2) time algorithms for the problems of covering a horizontally convex orthogonal polygon with the minimum number of orthogonal convex polygons and with the minimum number of orthogonal star-shaped polygons.

New results about the approximation behavior of the greedy triangulation

Abstract: In this paper it is shown that there is some constant c, such that for any polygon, with or without holes, with w concave vertices, the length of any greedy triangulation of the polygon is not long

Flexible bulk containers

Abstract: A flexible bulk container comprises a tubular side wall structure (1 to 4) of woven fabric, lifting means (13) at an upper end of the side wall structure, and a base (14) closing the lower end of the side wall structure. The base is in the form of a polygon having an even number of sides and comprising a plurality of thicknesses of said woven fabric. Each thickness is formed by two joined flaps of woven fabric forming integral extensions of the side wall structure and each extending from one side of the polygon towards the opposed side thereof. In each thickness of the base, each flap is of substantially right-angled triangle shape having a first adjacent side lying along one side of the polygon and a second adjacent side extending at right angles from said one side to the opposed side of the polygon, and the two flaps are secured together substantially along the hypotenuses thereof.

Polygon-filling apparatus used in a scanning display unit and method of filling the same

Abstract: A polygon-filling apparatus discriminates whether the polygon marks off ony one continuous lot of each scanning line according to a number of maximal values and a number of minimal values in a direction perpendicular to a scanning line and which transforms polygons discriminated as marking off two or more continuous lots on each scanning line to obtain only polygons marking off one continuous lot of each scanning line. The apparatus detects other apexes from any apex of the obtained polygon corresponding to maximum and minimum coordinate values in a direction perpendicular to the scanning line and paints a predetermined region according to ridge data on the left and right sides of each scanning line. The ridge data are obtained by interpolation between data indicative of neighoring apexes chained from apexes having maximum and minimum values.

Reachable grasps on a polygon: The convex rope algorithm

Abstract: An algorithm that finds the externally visible vertices of a polygon is described. This algorithm generates a new geometric construction, termed the convex ropes of each visible vertex. The convex ropes give the range of angles from which each vertex is visible, and they give all the pairs of vertices which are reachable by a straight robot finger. All of the convex ropes can be found in expected time order n, where n is the number of vertices of the polygon. We discuss the application of this geometric construction to automated grasp planning. The algorithm may also be useful in image interpretation and graphics where efficient computation of visible points is important. The direct application of the algorithm is restricted to two dimension since sequential ordering of vertices is required. Extension to three dimension would rely on well chosen intersecting or projective planes.

Image buffer having logic-enhanced pixel memory cells and method for setting values therein

Abstract: The system provides a relatively inexpensive raster-scan type graphics system capable of real time operation, utilizing logic-enhanced pixels within an image buffer, permitting parallel (simultaneous) calculations at every pixel. A typical implementation would be as custom VLSI chips. Each cell of the image buffer corresponds to a pixel of the display, and a processor at each cell enables calculation of the pixel color and the like for each polygon in the image covering that same pixel (cell) of the display. In the sequence of most general applications, each polygon is operated upon in sequence, and the image is built up as the polygons are processed without the necessity of sorting. With respect to each successive polygon, the following operations are effected: (1) all pixels within the polygon are identified; (2) the respective pixels which would be visible to the observer, that is, not obstructed by some previously processed polygon, are determined; and (3) the proper color intensities for each visible pixel are determined.

Tarski and Geometry

Abstract: Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.

Finding the minimum visible vertex distance between two non-intersecting simple polygons

Abstract: In this paper, we present an O(n log n) algorithm for finding the minimum Euclidean visible vertex distance between two nonintersecting simple polygons, where n is the number of vertices in a polygon. The algorithm is based on applying a divide and conquer method to two preprocessed facing boundaries of the polygons. We also derive an O(n log n) algorithm for finding a minimum sequence of separating line segments between two nonintersecting polygons.

Apparatus and methodology for automated filling of complex polygons

Abstract: A graphics co-processor that is autonomously responsive to an instruction for the filling of a complex polygon, as defined by an enumeration of P vertices is described. The co-processor preferably includes a micro-engine sequencer and ALU (arithmetic logic unit) for selecting a first vertex from the enumeration of P vertices and for decomposing the complex polygon into a set of P-2 triangles, wherein each triangle includes the first vertex and to successive vertices as presented in the enumeration of P vertices is derived. A sense value is derived for each of the resultant P-2 triangles and each triangle is filled with a predetermined fill quantity that is qualified by the respectively associated sense value of the triangle being filled. Thus, the present invention provides for the autonomous execution of a fill polygon instruction for polygons having such complexities as concavities, self-intersections, overlapping sections and "holes".

Dynamic C-oriented polygonal intersection searching

Abstract: A set of polygons is called c -oriented if the edges of all polygons are oriented in a constant number of previously defined directions. The intersection searching problem is studied for such objects, namely: Given a set of c -oriented polygons P and a c -oriented query polygon q , find all polygons in P that intersect q . It is shown that this problem can be solved in O (log 2 n + t ) time with O ( n log n ) space and O ( n log 2 n ) preprocessing, where n is the cardinality of P and t the number of answers to a query. Furthermore, the solution is extended to the cases in which P is a semidynamic or dynamic set of polygons. Whereas planar intersection searching can be carried out more efficiently for orthogonal objects (e.g., rectangles) it is expensive for arbitrary polygons. This suggests that the c -oriented solution be used in appropriate areas of application, for instance, in VLSI-design.

Tactile recognition by probing: identifying a polygon on a plane

Abstract: An outstanding problem in model-based recognition of objects by robot systems is how the system should proceed when the acquired data are insufficient to identify uniquely the model instance and model pose that best interpret the object In this paper, we consider the situation in which some tactile data about the object are already available, but can be ambiguously interpreted The problem is thus to acquire and process new tactile data in a sequential and efficient manner, so that the object can be recognised and its location and orientation determined An object model, in this initial analysis of the problem, is a polygon located on a plane; the case of planar objects presents some interesting problems, and is also an important prelude to recognition of three-dimensional (polyhedral) objects

Polygon mirror construction

Abstract: In a polygon mirror construction comprising a polygon mirror with a hole formed at the central position for passing through a shaft of a motor and a supporting means fixed to the shaft of the motor for supporting the polygon mirror, and a space, a groove or a heat insulating plate is formed between the polygon mirror and the supporting means, thereby a heat rise of the polygon mirror is prevented and high accuracy of the polygon mirror construction is assured.