Showing papers on "Point (geometry) published in 1985"
TL;DR: In this paper, a new methodology for describing the internal structure of planar point sets is proposed, which is based on parameterized measures of neighbourliness and gives rise to a spectrum of possible internal shapes.
Abstract: This paper outlines a new methodology for describing the “internal structure” (or “skeleton”) of planar point sets. The methodology, which is based on parameterized measures of neighbourliness, gives rise to a spectrum of possible internal shapes. Applications to the analysis of both point set and network patterns are described.
305 citations
01 Jan 1985
135 citations
TL;DR: An algorithm is presented for ray tracing generalized cylinders, that is, objects defined by sweeping a two-dimensional contour along a three-dimensional trajectory, which is possible to obtain highly realistic images for a very broad class of objects.
Abstract: An algorithm is presented for ray tracing generalized cylinders, that is, objects defined by sweeping a two-dimensional contour along a three-dimensional trajectory. The contour can be any 'well-behaved' curve in the sense that it is continuous, and that the points where the tangent is horizontal or vertical can be determined, the trajectory can be any spline curve. First a definition is given of generalized cylinders in terms of the Frenet frame of the trajectory. Then the main problem in ray tracing these objects, the computation of the intersection points with a ray, is reduced to the problem of intersecting two two-dimensional curves. This problem is solved by a subdivision algorithm. The three-dimensional normal at the intersection point closest to the eye point, necessary to perform shading, is obtained by transforming the two-dimensional normal at the corresponding intersection point of the two two-dimensional curves. In this way it is possible to obtain highly realistic images for a very broad class of objects.
69 citations
TL;DR: This paper presents a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history, and shows that with O ( n 2 ) preprocessing, it is possible to determine in O (log 2 n ) time which of n given points in E 3 is closest to an arbitrary query point.
Abstract: This paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times. We present a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history. The novelty of the method is to allow the set S to be only partially ordered with respect to queries and the time measure to be multi-dimensional. The generality of the method makes it useful for a number of problems in 3-dimensional geometry. For example, we are able to give fast algorithms for locating a point in a 3-dimensional complex, using linear space, or for finding which of n given points is closest to a query plane. Using a simpler, yet conceptually similar technique, we show that with O ( n 2 ) preprocessing, it is possible to determine in O (log 2 n ) time which of n given points in E 3 is closest to an arbitrary query point.
66 citations
01 Apr 1985-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: It is shown that digital straight lines (according to grid intersection digitizations) are special digital curves which may be uniquely recognized by m −1 projections into the 2-dimensional point grid Z2.
Abstract: A wide variety of algorithms and methods in computer graphics and digital image processing is based on point grids Zm defined by regular orthogonal grids in m-dimensional real space Rm, and on the metrics that may be defined on Zm, for m ≥ 2. In this paper, half-norms ∥ · ∥l, metrics δl, and point products 〈·, ·〉l are introduced characterizing different m-dimensional metric grid point spaces, for 0 <- l < m. Furthermore, grid point ditizations in m-dimensional space are defined and grid intersection digitizations for hyperplanes are analyzed. It is shown that digital straight lines (according to grid intersection digitizations) are special digital curves which may be uniquely recognized by m −1 projections into the 2-dimensional point grid Z2.
35 citations
TL;DR: By use of geometric transforms, the post-office problem for a finite set of points in two dimensions and certain two-dimensional point location problems are reduced to the former problem and thus also optimally solved.
34 citations
TL;DR: In this paper, it was shown that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions.
Abstract: Charles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.
32 citations
31 citations
01 Jan 1985
TL;DR: A regular decomposition of the space is used, leading to a tree structure, which makes it easy to detect intersections of 3D objects, which is a useful property in such applications as interactive design of three dimensional shapes.
Abstract: Indexing methods are very important for rapid processing of a large amount of data. In this paper we discuss a spatial index, that is, a method for indexing a three dimensional space. We use a regular decomposition of the space, leading to a tree structure. The advantage of a space decomposition method over storing data in the form of a table is the quick access to a point in question by using a leaf node as an index. A set of basic algorithms is presented for generation and modification of objects. This set makes it easy to detect intersections of 3D objects, which is a useful property in such applications as interactive design of three dimensional shapes.
31 citations
TL;DR: In this paper, an elastic, spring-supported beam along which a point mass is moving, is considered, and special features of the behaviour of such systems are pointed out, and the stability of the system point mass-beam is studied.
01 Jan 1985
TL;DR: In this paper, the authors consider a domain in iR which is locally convex at each point of its boundary except one, where the radial limits of f at (0, 0) from all directions in Ω exist and the restriction of f to Π η Η is a classical solution to the Dirichlet problem which continuously takes on all its boundary values.
Abstract: Let Ω be a domain in iR which is locally convex at each point of its boundary except one, say (0,0), Ψ be a continuous function of 9Ω, and f be the unique generalized (variational) solution of the minimal surface equation with boundary values Φ. Assume that f e C°(Ώ\{(0,0)}). Then the radial limits of f at (0,0) from all directions in Ω exist and the restriction of f to Π η Η is a classical solution to the Dirichlet problem which continuously takes on all of its boundary values, where Η is a closed half space with (0,0) e 3H. 1980 AMS Subject Classification. Primary 35J65, 35B99.
01 Jan 1985
TL;DR: In this article, a coordinate system for analytic geometry in 3D space was introduced, in which triples (a,b, c) of real numbers are considered, and the set of all such triples of real number is called the 3D number space.
Abstract: Analytic geometry in three dimensions makes essential use of coordinate systems. To introduce a coordinate system, we consider triples (a,b, c) of real numbers, and we call the set of all such triples of real numbers the three-dimensional number space. We denote this space by R’. Each individual triple is a point in R 3 . The three elements in each number triple are called its coordinates. We now show how three-dimensional number space may be represented on a geometric or Euclidean three-dimensional space.
TL;DR: In this article, a method for the numerical computation of a double turning point corresponding to a cusp catastrophe of a nonlinear operator equation depending on two parameters is developed. But this method is limited to the finite-dimensional case.
Abstract: A method is developed for the numerical computation of a double turning point corresponding to a cusp catastrophe of a nonlinear operator equation depending on two parameters. An augmented system containing the original equation is introduced, for which the cusp point is an isolated solution. An efficient implementation of Newton's method in the finite-dimensional case is presented. Results are given for some chemical engineering problems and this direct method is compared with some other techniques to locate cusp points.
TL;DR: The problem of preprocessing P so that for any query point q, the points of P in C+q can be retrieved efficiently is studied, and the existence of an optimal solution is demonstrated.
01 Dec 1985
TL;DR: Three special cases of increasing difficulty and generality of the hidden line elimination problem are studied, and applying some methods from computational geometry these problems can be solved with better worst-case bounds than those of the best known algorithms for the general problem.
Abstract: Hidden line elimination is a well-known problem in computer graphics and many practical solutions have been proposed. Only recently the problem has been studied from a theoretical point of view, taking asymptotic worst-case time- and spacebounds into account. Here we study three special cases of increasing difficulty and generality of the hidden line elimination problem. Applying some methods from computational geometry these problems can be solved with better worst-case bounds than those of the best known algorithms for the general problem.
Journal Article•
TL;DR: In this article, point sets everywhere dense in the plane have been used for point set construction and point set detection in space and time, using point sets from point sets in the air.
Abstract: Keywords: point sets everywhere dense in the plane Note: Professor Pach's number: [031] Reference DCG-ARTICLE-1985-001 Record created on 2008-11-14, modified on 2017-05-12
TL;DR: A survey of the present state of the generating functional which makes it possible to effectively study distributions of point random measures on a complete, separable metric space can be found in this article, where the principal attention is devoted to the study of distributions of configurations of infinite systems of statistical physics.
Abstract: This is a survey of the present state of the method of the generating functional which makes it possible to effectively study distributions of point random measures on a complete, separable metric space. The principal attention is devoted to the study of distributions of configurations of infinite systems of statistical physics — Gibbs distributions.
15 Jul 1985
TL;DR: If constant time suffices for deciding the inclusion of a point in C, the existence of an optimal solution is demonstrated: the algorithm requires O(n) space and O(k + log n) time for a query with output size k.
Abstract: Let P be a set of n points in the Euclidean plane and let C be a convex figure. We study the problem of preprocessing P so that for any query point q, the points of P in C + q can be retrieved efficiently. If constant time suffices for deciding the inclusion of a point in C, we then demonstrate the existence of an optimal solution: the algorithm requires O(n) space and O(k + log n) time for a query with output size k. If C is a disk, the problem becomes the well-known fized radius neighbor problem, to which we thus provide the first known optimal solution.
TL;DR: This note describes an improved version of the Nordbeck and Rystedt algorithm for determining whether a point is inside a polygon of arbitrary shape that is compatible with FORTRAN 77.
Patent•
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23 Jan 1985
TL;DR: In this article, the shape line information of a drawing can be outputted as vector information separately from the vector information of the other additional line information or the like, by forming a tracking start line detecting part and a shape line tracking part.
Abstract: PURPOSE: To separate the shape line information of a drawing and other additional line information or the like easily as vector information and to output the separated information to a computer by forming a tracking start line detecting part and a shape line tracking part to track the shape line. CONSTITUTION: Picture image data inputted from a picture image input part 1 to a picture image memory part 2 are converted into point sequence data by a point sequencing part 3 and stored in a point sequence data storing part. The tracking start line detecting part 6 displays the point sequence data on a display and the point sequence data is converted into a start segment for shape line tracking by specifying the shape segment with a light pen or the like from an operator to detect the shape segment. Consequently, the shape line tracking part 7 starts the tracking of the shape line connected to the start segment. In the shape line tracking method, a segment having the highest continuousness to the start segment or the segment already tracked as the shape line is adopted as the shape line at the crossing point of the shape line with another additional line or the like. Consequently, the shape line information of the drawing can be outputted as vector information separately from the vector information of the other additional line information or the like. COPYRIGHT: (C)1986,JPO&Japio
TL;DR: In this article, the authors point out an erroneous result appearing in the literature on the optimal design centering, tolerancing, and tuning problem and suggest alternate approaches to the solution of this problem, including a special-purpose subalgorithm.
Abstract: In this note, we point out an erroneous result appearing in the literature on the optimal design centering, tolerancing, and tuning problem. Using the same framework of outer approximations, we then suggest alternate approaches to the solution of this problem, including a special-purpose subalgorithm.