General Geometric Problem Research Articles

Семейства изолиний в пространстве с прямоугольной метрикой

The paper is devoted to planar linear sets generated by linear conditions which are realized for rectangular or Manhattan metrics. The aim of this paper is to give an approach to researching the properties of linear conditions defined on Manhattan plane. Linear conditions are written as a finite sum of products Manhattan distances and real numerical factors. The paper presents a constructive method to solve the next general geometric problem. Let finite sets of linear figures (segments of line, polygons) are given on Manhattan plane at general positions. Find a planar set corresponding to given linear condition based on the given sets. The set of broken lines corresponding to given conditions generates a family of isolines for given sets. Constructive algorithm to build the family is suggested. The algorithm is based on the rectangular Hanan lattice and calculation numerical values in each cell. Some theorems applied to these cases are formulated in the paper.

Образы линейных условий на плоскости с прямоугольной метрикой

In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.

Octahedral and Euclidean projections of a point to a linear manifold

Many applied problems reduce to the general geometric problem of finding a point of a linear manifold in a finite-dimensional space that is closest to the origin. There are many specific formulations of this problem, including the search for octahedral and Euclidean projections, i.e., vectors of a linear manifold with smallest octahedral and Euclidean norms. We consider the properties of solutions to the problem of finding points of linear manifolds that are closest to the origin and relations between these solutions under various specifications of the problem. In particular, we study the properties of octahedral and Euclidean projections and analyze the influence on these projections of variation of weight coefficients in the norms.

S-Metric Calculation by Considering Dominated Hypervolume as Klee's Measure Problem

The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a fast determination of the S-metric value is of essential importance. This work describes how to consider the S-metric as a special case of a more general geometric problem called Klee's measure problem (KMP). For KMP, an algorithm exists with runtime O(n log n + n(d/2) log n), for n points of d >or= 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications realize the algorithm without complex data structures and establish an upper bound of O(n(d/2) log n) for the S-metric calculation for d >or= 3. The performance of the new algorithm is studied in comparison to another state of the art algorithm on a set of academic test functions.

Shape optimization of three-dimensional folded-plate structures

The basic features of a previously described shape optimization capability included a general geometric problem description format and adaptive finite element analysis. The finite element mesh for each design step was created from the boundary information only and refined using finite element solution results. At that time, only structures in a single plane could be treated. This work has now been extended to handle structures created from an assembly of segments that may be rotated into any plane as well as to contain the nonplanar surface curvature within any segment. In order to handle these more general problems, it was necessary to introduce a more efficient mesh generation scheme. Several design examples are presented to demonstrate each of the features.