Basic Geometric Elements Research Articles

Observation of fivefold symmetry structures in computer models of dense packing of hard spheres

Structures of fivefold symmetry were obtained in computer models of dense packings of hard spheres containing crystalline regions (η>0.639). Such structures are known to exist in small particles and thin films; however our models are specimens of a rather bulk phase (10000 spheres in a box with periodic boundary conditions). This observation indicates that the fivefold structures can also exist in real bulk systems and play a role in the process of homogeneous crystallization of simple liquids and ageing of amorphous solids. The Voronoi–Delaunay approach is used for disclosing these structures. The Delaunay simplexes are the basic geometrical elements for this analysis. Having a quantitative measure for the shape of the simplexes, one can mark (color) Voronoi sites according to a given physical criterion to reveal aggregates of atoms with a given structure. Aggregates consisting of good tetrahedral simplexes are studied in this work.

An object-oriented data model for complex objects in three-dimensional geographical information systems

Developing a three-dimensional (3D) data model for Geographic Information Systems (GIS) is an essential and complex issue. 3D modelling in GIS is becoming ever more important for the development of cyber cities and digital earth, which have recently become feasible. A competent 3D model forms an efficient foundation for 3D visualization, query and spatial analysis. As a development of the existing 3D models, this study proposes particular improvements in handling complex 3D objects. We present an object-oriented data model for handling complex 3D objects in GIS. First, the conceptual data model is developed based on the principle of object-oriented (OO) data modelling. This model is designed based on the following three basic geometric elements: node, segment and triangle. Accordingly, the abstract geometric objects are defined: including points, lines, surfaces and volumes. Second, the corresponding 3D logical model is designed based on the defined abstract objects and the relationships between them. Third, a formal representation of the 3D spatial objects is described in detail. Fourth, a prototype 3D GIS is developed based on the proposed 3D data model. Finally, we describe the results of an experimental study to reconstruct 3D objects using this 3D GIS and a comparison with the performance of other 3D data models. The proposed model is able to handle complex objects, such as complex buildings and TV towers, which is an essential functionality for building large-scale cyber cities, such as for Hong Kong. The proposed data model proves to be very efficient, particularly in visualization and rendering. The experimental results show which the data volume of the proposed model is compacted and the visualization speed for 3D objects is improved, compared with the existing models.

A visual analysis of anthropomorphism in the Kabbalah: dissecting the Hebrew alphabet and Sephirotic diagram

A complex series of mystical texts and rituals within medieval Judaism, the Kabbalah endeavors to explain God by contemplating the anthropomorphic traits of designed artefacts - the Hebrew letterforms and the Sephirotic diagram. This article examines anthropomorphism in these artefacts from a design point of view. The author focuses on the relationships among visual elements within an artefact, and connections between the artefact’s formal qualities and the form of depictions of the body. Despite the Hebraic fear of graven images, the visual form of the Sephirotic diagram can be understood fundamentally as a nuanced, abstract representation of the human body. The designers of this diagram incorporate basic geometric elements of the body and appropriate the visual vocabulary of medieval anatomy. Interpreted visually, the Hebrew alphabetic ‘characters’ look and act like living creatures - they are fleshy beings wielding immense creative ability. Such design-based interpretations of the visual form of the Hebrew letters and the Sephirot evoke new and exciting anthropomorphic renderings.

Scale‐Up in the Design of Extrusion Dies

AbstractThe similarity theory and the scale‐up procedure developed from it make it possible to convert a technical‐physical process from one scale to another – generally in the upwards direction. In the design of extrusion dies with an annular die gap, the die is broken down into basic geometric elements, and scale‐up rules are then drawn up for the individual elements. In this project, scale‐up rules were found that satisfy both full and partial similarity. The pseudoplasticity and elasticity of the melt were also taken into consideration. A comparison of the model laws obtained with the results of finite element calculations showed good conformity between the models and the production versions. Deviation in pressure difference between the model and the production version assuming flow behavior in accordance with the power law (Di = const. and L = const.).magnified imageDeviation in pressure difference between the model and the production version assuming flow behavior in accordance with the power law (Di = const. and L = const.).

A preliminary study of a deformable body model for clothing simulation

The body model which has been utilized in our clothing simulation does not deform and gives a boundary condition for mechanical calculation. To determine the shape of clothing in the case where clothing and body mechanically interact with each other, the body model used for this purpose has to be deformable. In this study, basic techniques for realization of the deformable body model were investigated. A tetrahedron was defined as a fundamental element for mechanical calculation of solid, and it was formulated with ordinal strain. Four kinds of cubes consisting of six tetrahedrons were defined as basic geometrical elements for constructing solids. Two kinds of cantilevers were constructed from the cubes and mechanical simulation was carried out with proper mechanical properties. A method of estimating internal mechanical properties of the human body was tested. The method is a modification of the simulation and is one of inverse problems. Treatment of collision is required for the simulation in which clothing and body mechanically interact with each other. The treatment of collision is based on a triangular element, and the processes consist of its detection and resolution. Simulation of a right cylinder solid wound by fabric like pipe was carried out to check collision treatment.

Exploring Isovist Fields: Space and Shape in Architectural and Urban Morphology

The space that can be seen from any vantage point is called an isovist and the set of such spaces forms a visual field whose extent defines different isovist fields based on different geometric properties. I suggest that our perceptions of moving within such fields might be related to these geometric properties. I begin with a formal representation of isovists and their fields, introducing simple geometric measures based on distance, area, perimeter, compactness, and convexity. I suggest a feasible computational scheme for measuring such fields, and illustrate how we can visualize their spatial and statistical properties by using maps and frequency distributions. I argue that the classification of fields based on these measures must be a prerequisite to the proper analysis of architectural and urban morphologies. To this end, I present two hypothetical examples based on simple geometries and three real examples based on London's Tate Gallery, Regent Street, and the centre of the English town of Wolverhampton. Although such morphologies can often be understood in terms of basic geometrical elements such as corridors, streets, rooms, and squares, isovist analysis suggests that visual fields have their own form which results from the interaction of geometry and movement. To illustrate how such analysis can be used, I outline methods of partitioning space, covering it with a small number of relatively independent isovists, and perceiving space by recording properties of the isovist fields associated with paths through that space.

Constraint-based Object Modelling

A two-dimensional constraint-based geometric modelling system FLEXI is presented. Constraints are solved by local propagation of known states. The weakness of local propagation ( it cannot solve cyclic constraints) is explained by an example. A biconnected constraint-description graph is used to support local propagation. As a result of constraint solving, an acyclic constraint-description graph is obtained and it can be observed as a parametric description of a geometric object. The acyclic constraint-description graph stores the sequence of applying constraints for the generation of instances. It is shown that the time complexity of the algorithm for generation of an instance is O(n⊃2⊃,), where n is the number of basic geometrical elements. In the last part of the paper, we demonstrate how geometrical relations among geometrical objects can be established by introduced geometrical constraints. Finally, FLEXI is appraised against criteria for an ideal constraint-based geometric modelling system.

Task-Specific Gauge for the Inspection of Coordinate Measuring Machines

Coordinate measuring machines (CMMs) frequently are used for measuring tasks at bevel gears even though probing at inclined tooth profiles causes enormous length measuring uncertainties. In order to analyse these faults on principle a master bevel gear can be used, which however is very difficult to manufacture and to calibrate in a sufficient accuracy. Therefore a task specific gauge consisting of basic geometric elements has been developed, which can be calibrated with smallest uncertainties. After the tested CMM has probed a grid of defined measuring points, the maximum quality of bevel gears to be measured can be assessed.

Relative position and its precision (Removing the influence of the datum)

Abstract Coordinates and their variance matrix resulting from a least squares adjustment are datum dependent. This dependence is seen by the deterioration in coordinate precision with increasing distance from the fixed points (datum). Coordinate differences (relative position) which may be alternatively expressed in the form of bearings and distances still depend on the datum through the selection of a reference direction. Thus a method for expressing relative position that is truly independent of any datum must use the basic geometrical elements of angles and distances. The example of a small horizontal network is used to show how coordinates and their variance matrix may be changed by selecting different datum points. The values and precision of angles and distances extracted from the different solutions, however, remain the same.

Shape matching using optimization techniques

Automated vision inspection systems are the most versatile non-contact inspection systems. They are fast, compact, and remarkably accurate, thereby making 100% on-line inspection possible in modern manufacturing systems. Roundness in one of the basic geometric elements in engineering design and manufacturing. As a result its inspection is an important part of the inspection of many industrial parts. This paper presents two least-squares techniques to determine the out-of-roundness error of circular production parts. This error is measured with respect to an ideal circle of specified radius.

The Optimization of Dimensions of an Adjustable Mechanism for a Packaging Machine

Mechanisms form the basic geometrical elements of many mechanical devices including automatic packaging machinery, typewriters, textile and printing machinery, and others. A mechanism typically is designed to create a desired motion of a rigid body relative to a reference member by the help of gears, cam systems or spatial linkages. In flow pack machines a tube of wrapper containing the products moves with a uniform velocity, while the reciprocating heads move forward and backwards sealing different products. In an existing machine, these motions are produced by a rather complex mechanism involving cams and adjustable links. The paper suggests replacing these cams by a suitable quick-return mechanism with a minimum number of adjustable links. The dimensions of this mechanism are optimized such that the motions produced are as near as possible to those obtained by the original cam mechanisms. The simplification can result in reduction in the mass of different components and existing forces in the mechanism.

Automated inspection of circular parts

Automated vision inspection systems are the most versatile non-contact inspection systems. They are fast, compact, and remarkably accurate, thereby making 100% on-line inspection possible in modern manufacturing systems. Roundness in one of the basic geometric elements in engineering design and manufacturing. As a result its inspection is an important part of the inspection of many industrial parts. This paper discusses least-squares and minimax mathematical models and algorithms to determine the out-of-roundness error in automated roundness inspection. These errors are measured with respect to an ideal circle of specified radius.