Showing papers on "Point (geometry) published in 1974"

Three-dimensional reconstruction from projections: a review of algorithms.

Abstract: Publisher Summary This chapter provides an overview of three-dimensional reconstruction from projections. The chapter reviews the algorithms that have been proposed to solve the reconstruction problem. The known reconstruction algorithms are classified into four categories—summation, the use of Fourier transform, analytic solution of the integral equations, and series expansion approaches. For each class of algorithms several points need to be considered—a general intuitive description, a precise mathematical description of a typical reconstruction method of the class, and a brief description of other methods in the class. All algorithms for reconstruction take as input the projection data, and all produce as output an estimate of the original structure based on the available data. The estimate varies from method to method. The relative performance of the various methods depends on the object and how the data are collected. The simplest algorithm for reconstruction is to estimate the density at a point by adding all the ray sums of the rays through that point. The Fourier method depends on transforming the projections into the Fourier space, where they define part of the Fourier transform of the whole object. Each projection may be shown to yield values on a central section of the Fourier space, which is a line or plane through the origin at an angle corresponding to the direction of the projection in real space.

Green'sFunction for Lamb'sProblem

Abstract: Summary The complete solution to the three-dimensional Lamb'sproblem, the problem of determining the elastic disturbance resulting from a point force in a half space, is derived using the Cagniard-de Hoop method. In addition, spatial derivatives of this solution with respect to both the source co-ordinates and the receiver co-ordinates are derived. The solutions are quite amenable to numerical calculations and a few results of such calculations are given.

Algorithm for a least-distance programming problem

Abstract: An algorithm is developed for the problem of finding the point of smallest Euclidean norm in the convex hull of a given finite point set in a Euclidean space, with particular attention paid to the description of the procedure in geometric terms.

Method and apparatus for localization of failures on electrical lines

Abstract: At a measuring point serving at least one adjacent line segment instantaneous values of current and voltage are measured in sequential pairs, either continuously or at intervals by delaying these values by discrete amounts of delay corresponding to twice the wave propagation time on the line from the measuring point to a number of selected reference (appraisal) points along the line segment, and then combining the delayed value pairs corresponding to each appraisal point with a real time value pair. A voltage value at each appraisal point at the mid-point of the delay period is determined by use of delayed measured values that take account of line propagation characteristics such as damping. In one form of processing, the appraisal point voltage thus obtained is then squared or otherwise processed to provide a unimodal function, and then further processed by an average function generator to yield an evaluation function for the appraisal point which can effectively be compared with one or more limit values for control of warning and cut-off devices designed to be activated in case of a failure. In a second form of processing, the gradients of the appraisal point voltage at the two ends of the line segment in question are continually or sequentially calculated and their respective signs are compared in such a way as to activate warning and cut-out systems when the presence of a voltage minimum within the line segment is detected. The system operates without circuits of the kind that require measured voltages and currents to be resolved into sinusoidal components or proper evaluation of transients.

Three-dimensional cinematography: a flexible approach.

Abstract: The purpose of this study was to develop a procedure for three-dimensional cinematography in which 2 or more cameras could be placed in any position with the only restrictions that a common origin point must be in the field of view and that the optical axes should intersect. Each desired point lies on a line defined by the nodal point of the lens and the projection point. The projection point lies on a plane which is parallel to the film plane, perpendicular to the optical axis, and contains the common origin point. Two lines so defined, 1 from each camera, theoretically intersect at the desired point. The positions of two 35-mm Nikon cameras with 50-mm Nikkor lenses were determined with the use of a transit and were relative to an arbitrary Cartesian coordinate system. To validate the procedure a 17-point coordinate grid, two 8-point concentric circles and the origin were drawn on the floor. A plump bob was suspended over each point with markers at each foot level up to 4 ft. A Vanguard film ana...

Fixed point theorems for point-to-set mappings

Abstract: Abstract Let (X, d) be a metric space. Let Xc be the family of all non-empty closed subsets of X endowed with the Hausdorff metric D induced by d. Let f be a non-expansive mapping of X into Xc (D(f(x), f(y))≤d(x, y) for all x, y in X). Some fixed point theorems are obtained by imposing certain conditions on f and X.

The solvability of the problem of the membership of a point in an object in a geometric data-processing language ☆

Abstract: A language for describing geometrical objects in three-dimensional space was presented in [I] . Geometrical objects of the type “point”, “line”, “ surface” and “body” were considered. The description consisted of geometrical rules for geometrical expressions. An object was regarded as defined, if for an arbitrary point of space it was possible to decide whether it belongs to this object or not. The algorithm for working out the answer to this question was regarded as the act of specific representation of the language.