Showing papers on "Euclidean distance published in 1980"
TL;DR: It is shown that skeletons can be produced by simple procedures and since these are based on Euclidean distances it is assumed that they are superior to skeletons based on d4−, d8−, and even octagonal metrics.
1,777 citations
216 citations
Journal Article•
TL;DR: In this article, the authors introduced a new affine invariant of finite dimensional normed spaces, called the volume ratio of a Minkowski space, which is defined as the ratio of the number of orthogonal subspaces of a normed space with respect to the suitable euclidean norm on the space.
Abstract: We introduce a new affine invariant of finite dimensional normed space. Using it we show that for some large classes of finite dimensional normed spaces there is a constant C such that every normed space X contains two subspaces El, E2, orthogonal with respect to the suitable euclidean norm on X and satisfying d(Ei, l2[dlm X/2]) s C. 0. Introduction The motivation of this paper is to investigate which finite dimensional normed spaces admit Kashin’s decomposition on nearly Euclidean orthogonal subspaces. Kashin discovered (cf. [7] and [ 11 ]) the following (K) For arbitrary positive integer n the space 12n contains two orthogonal (in the sense of l12n) subspaces E,, E2 satisfying The proof of (K) given in [11 ] depends only on the fact that where B(ln1) and B(l2) are the unit balls of In and I2 respectively. This suggests to investigate the volume ratio an affine invariant of Minkowski spaces. We define the volume ratio of a finite dimensional normed space X by 0010-437X/80/03/0367-19$00.20/0
91 citations
TL;DR: A combinatorial algorithm for the problem in three dimensions which maximizes the minimum Euclidean distance from a given set of points and is compared with existing nonconvex programming algorithms.
Abstract: The problem considered is to locate a point in a given convex polyhedron which maximizes the minimum Euclidean distance from a given set of points. The paper describes several possible application areas and shows the existence of a finite set of candidates for the optimal solution. A combinatorial algorithm is presented for the problem in three dimensions, and it is compared with existing nonconvex programming algorithms.
77 citations
TL;DR: For solving the Euclidean distance Weber problem Weiszfeld proposed an iterative method that can be applied to generalized Weber problems in Banach spaces and Fermat's principle in geometrical optics.
Abstract: For solving the Euclidean distance Weber problem Weiszfeld proposed an iterative method. This method can also be applied to generalized Weber problems in Banach spaces. Examples for generalized Weber problems are: minimal surfaces with obstacles, Fermat's principle in geometrical optics and brachistochrones with obstacles.
72 citations
TL;DR: In this paper, the problem of finding a point on the sphere S 2 = {x = (x, y, z)¦x2 + y2 + z2 = 1} which minimizes the weighted sum of the distances to N given destination points xj on S2 is studied.
Abstract: The problem of finding a point on the sphere S2 = {x = (x, y, z)¦x2 + y2 + z2 = 1} which minimizes the weighted sum of the distances to N given destination points xj on S2 is studied. Three different metrics are considered as distances between points on S2: (A), square of Euclidean distance; (B), Euclidean distance; (C), great circle distance. Non uniqueness of minimizers is demonstrated and some pathological cases are studied. An algorithm, analogous to the Weiszfeld algorithm for the classical unconstrained Weber problem is formulated, and its convergence properties are investigated. A necessary and sufficient condition for a destination point to be a local minimizer is derived. Finally, a modified form of Steffensen's acceleration is given and the results of numerical tests are presented. These results illustrate the predictions of the theory, and confirm the effectiveness of Steffensen's acceleration.
54 citations
TL;DR: A more unified and numerically stable approach for solving minisum and minimax location problems is proposed, and preliminary computational experience suggests that this approach compares favourably with other methods.
Abstract: There is a rapidly growing interdisciplinary interest in the application of location models to real life problems. Unfortunately, the current methods used to solve the most popular minisum and minimax location problems are computationally inadequate. A more unified and numerically stable approach for solving these problems is proposed. Detailed analysis is done for the linearly constrained Euclidean distance minisum problem for facilities located in a plane. Preliminary computational experience suggests that this approach compares favourably with other methods.
43 citations
TL;DR: Directed canonical analysis as discussed by the authors is an extension of the general form of canonical analysis, which is a method for reducing the dimensionality of multivariate data sets with minimum loss of discriminatory variance.
Abstract: Directed canonical analysis is presented as an extension of the general form of canonical analysis, which is a method for reducing the dimensionality of multivariate data sets with minimum loss of discriminatory variance. The reduction takes the form of a linear transformation, y = Cx, that condenses the discriminatory variance onto a relatively few, high-variance orthogonal discriminant axes. Canonical analysis is developed as an analog to the one-way MANOVA. The directed extension allows user-specified contrasts to define linear relationships that are known or suspected to exist within the data. The linear transformation C is defined by means of the symmetric canonical form of the matrix eigenproblem. Canonical and principal components transformations and various distance classifiers were applied to 3 representative remotely sensed MSS data sets. Results indicate that use of a piecewise maximum likelihood classifier with the directed canonical discriminant axes will give the best overall combination of classification accuracy and computational efficiency if adequate sample sizes are available to estimate category statistics. For small sample sizes, piecewise Euclidean distance with the general canonical axes is recommended. In canonically transformed space, Euclidean distance is equivalent to the Mahalanobis classifier.
25 citations
TL;DR: In this article, necessary conditions are established for a point contained in the interior or boundary of a convex feasible subset of Euclidean space to be quasi-undominated in an anonymous simple game.
Abstract: Necessary conditions are established for a point contained in the interior or boundary of a convex feasible subset of Euclidean space to be quasi-undominated in an anonymous simple game. Most of the conditions are behaviorally intuitive and imply pariwise symmetries among utility gradients.
25 citations
24 citations
TL;DR: In this paper, it was shown that although a Euclidean version of Israel's theorem does provide a type of uniqueness theorem for the EKG, it does not allow one to form conclusions about the uniqueness of the EKS solution.
Abstract: The Euclidean section of the classical Lorentzian black-hole solutions has been used in approximating the functional integral in the Euclidean path-integral approach to quantum gravity. In this paper the claim that classical black-hole uniqueness theorems apply to the Euclidean section is disproved. In particular, it is shown that although a Euclidean version of Israel's theorem does provide a type of uniqueness theorem for the Euclidean Schwarzschild solution, a Euclidean version of Robinson's theorem does not allow one to form conclusions about the uniqueness of the Euclidean Kerr solution. Despite the failure of uniqueness theorems, ''no-hair'' theorems are shown to exist. Implications are discussed. A precise mathematical statement of the Euclidean black-hole uniqueness conjecture is made and the proof, left as an unsolved problem in Riemannian geometry.
TL;DR: In this paper, a new formulation for the study of the asymptotic structure of a gravitational field at spatial infinity is presented, and the disadvantages of existing formulations are identified in order to recognize the underlying causes and exclude them from the new formulation.
Abstract: A new formulation for the study of the asymptotic structure of a gravitational field at spatial infinity is presented. First, the disadvantages of the existing formulations are identified in order to recognize the underlying causes and exclude them from the new formulation. It is concluded that neither conformal nor projective completion should be used. From a study of the Euclidean space we obtain a method of completion of a three‐dimensional space (H,g) with positive‐definite metric g so that a two‐dimensional boundary L is attached to the space at infinity and a three‐dimensional positive definite C∞ metric ĝ exists near and on L. The whole method is based on replacing the conformal transformation of the conformal completion by the relation Ω−2gij−Ω−4gimgjnΩ ;mΩ;n=?ij−?im?jn Ω‖mΩ‖n. Thus the concept of asymptotic simplicity is defined. Then the additional conditions are determined for the space to be asymptotically Euclidean. The asymptotic symmetries and the uniqueness of the boundary are examined bri...
TL;DR: In this paper, a simple Green's formula for the Euclidean Dirac operator in Schwinger's real formalism allows us to study some localization properties of the Dirac field.
Abstract: A simple Green’s formula for the Euclidean Dirac operator in Schwinger’s real formalism allows us to study some localization properties of the Euclidean Dirac field A rather complete analysis follows of the connection between the Grassmann structure of the Euclidean field and the Clifford structure of the field at sharp time, analyzed in terms of its independent degrees of freedom
TL;DR: A conditional binomial probability model is demonstrated and related to the distance metric and the use of equal frequency classes for characters can overcome problems and validate the assumptions necessary for the probabilistic interpretation of distance values.
Abstract: Measures of resemblance are generally inadequate in handling within-operational taxonomic unit (OTU) character variability. A new and generalized Euclidean distance metric is related to existing association coefficients and can be isotonic with commonly applied distance measures. Distortions of the distance matrix arise as a result of distributions of data, properties of special types of characters, and intra-OTU variability. The use of equal frequency classes for characters can overcome these problems and validate the assumptions necessary for the probabilistic interpretation of distance values. A conditional binomial probability model is demonstrated and related to the distance metric.
TL;DR: With these decision-theoretic classifiers using discretized ECG features the fall in classification accuracy as a result of wrong representation of one feature rarely exceeds 20% and is usually less than 10%.
TL;DR: A variant of Simmons' argument is used to find some analogous estimates in higher dimensional Euclidean space of two-colored regions if r=<2g-2 so that rg<2.
TL;DR: In this paper, the determination of a large class of regular and singular Euclidean self-dual Yang-Mills field configurations is reduced to the solution of a set of linear algebraic equations.
Abstract: The determination of a large class of regular and singular Euclidean self-dual Yang-Mills field configurations is reduced to the solution of a set of linear algebraic equations. The matrix of the coefficients is a polynomial functions ofx and the rules for its construction are elementary.
TL;DR: In this article, a Euclidean metric for colorimetric evaluations based on an opponent-color system was described, which was shown to work well for predicting the results of the standard color-discrimination experiment.
Abstract: A Euclidean metric for colorimetric evaluations based on an opponent-color system was described by Ingling and Tsou. The metric was shown to work well for predicting the results of the standard color-discrimination experiment. The metric is applied to the surface-color data of the Munsell collection, the Morley, Munn, and Billmeyer data, and the data from the ISCC Committee on Color Difference Problems.
TL;DR: A computer-aided verification of a locus result which shows the impossibility of finite universal axiomatizability for the case of the city-block metric in R 3 .
TL;DR: In this paper, a general canonical form for elements in a ring Euclidean with respect to a real valuation is established, and it is shown that this form is unique and minimal thus gives the arithmetical weight of an element with respectto a radix.
Abstract: In this paper a general canonical form for elements in a ring Euclidean with respect to a real valuation is established. It is also shown that this form is unique and minimal thus gives the arithmetical weight of an element with respect to a radix.