Constructive Geometry Research Articles

O nouă abordare a problemei trisecției unghiului

The article presents new formulas in the problem of trisection of the angle with the help of conics and thanks to this fact a new vision is proposed for solving the problem of trisection of the angle with the ungraded ruler and the compass, using the ellipse drawn in the plane, not solved so far. Currently, the trisection of the angle with the help of the parabola (R. Decarts) and with the help of the hyperbola (M. Chasles) are described in the mathematical literature. The trisection of the angle with the help of the ellipse drawn in the plane is described in symbols and textually since not all readers know the symbols used in constructive geometry. The content of this article contributes to solving the angle trisection problem with the ungraded ruler and the compass. The information in the article is of interest in the field of constructive geometry.

Numerical modelling for the seismic assessment of complex masonry heritage buildings: the case study of the Giralda tower

Existing heritage buildings can be especially vulnerable to earthquakes. They were designed only considering gravity loads and some of them are located in earthquake prone areas, such as the southwestern Iberian Peninsula. Besides, there is a high uncertainty in the definition of their constructive parameters and complex geometry. Due to that, it is paramount to develop accurate numerical models to obtain a reliable assessment of their seismic behaviour. Given this, the main objective of this manuscript is to analyse the seismic behaviour of the Giralda tower, located in Seville (Spain). It was declared as a Word Heritage Site of Outstanding Universal Value by the UNESCO in 1987. Seville has a moderate seismic hazard, but it has been shown that the soft alluvial strata amplify the seismic action. The tower has a brick and stone masonry structure, which was constructed in several construction phases. A 3D Finite Element Model of the tower has been developed using OpenSees software, employing a 3D CAD model. Modal analyses and nonlinear static analyses have been applied to calibrate and to assess the tower’s seismic behaviour. The results showed significant differences in function of the load pattern. It should be remarked that the boundary conditions have a notable effect upon getting a good calibration of the model. Regarding the damage, it has been found to match the historic records: the ramps would be ruined and, in the outer wall, it would be concentrated near the openings, especially close to the belfry.

CONSTRUCTIVE GEOMETRY IN IMPLEMENTATIONS OF MODERN 3D GRAPHICS

3D graphics are one of the crucial development trends of modern digital technologies. Engineering and manufacturing, architecture, design, cinematography, education, and the game industry are an incomplete list of industries where it is actively used. Specialists in 3D graphics are in high demand in the labor market. Their proper training presupposes high-quality knowledge of geometrical sciences, in particular – constructive geometry. Note that constructive geometry is an integral part of modern school mathematics education. That is why, even in the conditions of the school, the teacher should skillfully apply the demonstration capabilities of three-dimensional graphics. It will also encourage students of a comprehensive school to apply knowledge of constructive geometry in practice in the area of 3D modeling. This approach will make it possible to demonstrate the importance and interconnectedness of knowledge in geometry and computer science. Therefore, the article reveals the importance of interdisciplinary connections between the specified disciplines in the context of research, demonstration, and application aspects. In particular, the nuances of using the GeoGebra dynamic geometry complex for conducting computational experiments and creating spatial models based on tasks from a school spatial geometry course are described. After all, modern capabilities of software tools make it possible to demonstrate in real time all the transformations that took place during drawing modeling on the picture plane. The importance of the applied value of constructive geometry for 3D modeling reveals based on examples of solid and polygonal modeling of virtual spatial objects. In particular, the steps of creating a solid model of a pyramid, which is formed by cutting it off with a plane from a regular quadrangular pyramid, are illustrated by the basis of calculations and constructions, which are performed using techniques of constructive geometry. All stages are described and done using the TinkerCAD online modeling service tools. An example of using the Blender program for creating polygonal 3D models is also provided. In particular, the significant aspects of the part modeling process are presented in the example of a task from a drawing textbook. The importance of planimetric constructions in the process of performing high-precision polygonal modeling is also emphasized. The article contains many figures that illustrate the essential stages of modeling. The materials presented can be used to prepare lessons in either mathematics or computer science and can be used to conduct integrated classes that draw on both subjects. Possible prospects for further research on this topic are also presented.

G1 interpolation of v-asymmetric data with arc-length constraints by Pythagorean-hodograph cubic splines

The interpolation problem of a sequence of points with prescribed segment arc-length by PH cubic spline is addressed. It turns out to be a v-asymmetric G1 Hermite interpolation problem with specified arc length. Using rectifying control polygon and reduction to a canonical form, it is shown that the problem can be expressed in terms of finding the real solutions to a system of nonlinear equations in two variables. A detailed and thorough analysis of the resulting system of nonlinear equations and closed form solutions are provided for any given data. It is confirmed that this construction of G1 Hermite interpolants of specified arc length admits at least two formal solutions, both of which have attractive shape properties and some (if any) must be discarded due to undesired looping behavior. The algorithm developed herein offers simple and efficient closed-form solutions to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. Moreover, the algorithm is illustrated by several numerical examples.

Rhythm and Architecture in Prague around 1400: Changing Architectural Paradigms at St. Vitus’s Lodge

In this contribution I focus on the relationship between rhythm, architecture, and music, especially around 1400. Medieval architecture and music are both grounded in the classical theory of proportion. In the sequence of the seven artes liberals, arithmetic and geometry-the arts associated with architecture-are placed just before music, forming the quadrivium along with astronomy. Proportion, order, and harmony play an essential role in both music and architecture. In medieval architecture, rhythm was the consequence of the use of classical theories of proportion with the application of constructive geometry, and is seen most clearly in monumental longitudinal spaces of basilicas where rows of piers or columns and vaulting shafts corresponding to bays form regular rhythm in the interior to which the same rhythm of windows and buttresses or flying buttresses corresponds on the exterior. The paradigm changes only when this regular rhythm is loosened and connected with a less rigid treatment of the existing tradition of cathedral construction as can be seen in St. Vitus’ Lodge in Prague, a project led by Peter Parler. In the article I visually present this moment of radically changed rhythm and order at the end of the fourteenth century in Prague and compare it to musical developments at that time.

Symmetric Tangling of Honeycomb Networks

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.

Surface Quality Optimization in Micromachining with Cutting Tool Having Regular Constructive Geometry.

In this paper we studied the influence of micromachining parameters on processed surface quality. Usually in discussions about micro-cutting or micromachining, the grinding or diamond turning processes are considered. Cutting tools used in the mentioned processes do not have regular constructive geometry and, in this case, it is difficult to use constructive geometric parameters such as clearance angle α or rake angle γ to optimize the quality of the machined surface. In order to determine the influence of the cutting tool’s constructive geometry on the hardness of the machined material, we used a fractional factorial design of a centered and rotatable type 26−1. A mathematical model based on five independent cutting parameters was created that allowed optimization of surface quality based on obtained roughness. The results can be applied in micromilling or microturning.

Device Substantiation for Generating Artificial Rain Drops by Pneumohydraulic Liquid Spraying

The authors studied the pneumohydraulic device indicators for spraying liquids for irrigation, nutrition and protection of agricultural plants, taking into account the principles of water and energy conservation, based on preliminary gas saturation of sprayed water and the use of a cavitation effect in the design of the aerator unit during ejection and supply of air under pressure. (Research purpose) To determine the technological parameters of a pneumohydraulic device for spraying liquids to obtain controlled dispersive artificial rain and substantiate the choice of its optimal technical parameters depending on the operating modes. (Materials and methods) The authors used an algorithm for calculating parameters in EXCEL or WPS spreadsheet processor and mathematical expressions. (Results and discussion) The authors theoretically determined the minimum and maximum calculated parameters of the constructive solution geometry for spraying the liquid phase: water nozzle, air nozzle channel, mixing cell, middle annular gap, outlet nozzle. They changed indicators of operating water pressure – 0.20; 0.25; 0.30 and 0.35 megapascals; air – 0.25 and 0.30 megapascals, provided the water flow rate from 0.002 to 0.010 liter per second and air – from 0.0005 to 0.0090 kilogram per second. With an increase in the water flow rate within the specified limits and the ejection coefficient from 0.5 to 0.9, a linear increase in the average annular gap diameter from 2 to 15 millimetres was revealed, as well as a nonlinear dependence of the increase in the sprayer mixing cell diameter from 5 to 20 millimetres. The authors showed the possibility of reducing the mixing cell diameter if the water pressure was increased from 0.25 to 0.35 megapascal's and the air pressure was from 0.20 to 0.30 megapascals. They obtained the parameters values for the designed and experimental samples development, which turned out to be significantly less than when operating in the air ejection mode: the outlet nozzle and the middle annular gap – by 16 percent, the air nozzle – by 23, the diameter of the mixing cell – by 50 percent or more. (Conclusions) The authors obtained calculated data to optimize technological parameters and design solutions, which would speed up the manufacture of designed and model samples of the device and its experimental testing for the generation of dispersive artificial rain drops.

Development and application of the geometry constructions language to building computer geometric models

Geometric models for industrial purposes are created in CAD systems making use of geometric constructions commands. At the same time, the so-called constructive or synthetic (as opposed to analytical) method to create a geometric model is implemented. The set of tools of any CAD system is limited, so accomplishing a specific task will require repeating a series of geometric constructions of the same type to form the desired shape. In this paper, a specific GC language (Geometry Constructions Language) is proposed to introduce an automated process of geometric modeling. To do this, theoretical issues of constructive geometry are analyzed, the main objects and syntax of the GC language are identified. The paper provides an example of building a model of a temple dome using GC language translator. It is noted that the GC language allows geometric modeling of regular curves (including trajectory of objects) and surfaces and applying new geometric correspondences and transformations. At the same time, geometric models in CAD systems are created faster since you don’t have to repeat manually the same type of construction or develop new equations and formulas for calculations.

Hardware and Software Implementation of Constructive Geometric Models

The article is devoted to the consideration of a number of issues of hardware and software implementation of constructive geometric models. A rich arsenal of theoretical research in the field of constructive geometry has not been properly used for a long time due to the lack of tools for translating such models using computer technology. The development and improvement of the Simplex geometric modeling system, in which any geometric design is considered as a converter of information represented by signals of a geometric nature, has opened the possibility of applying the achievements of geometric science in computing applications, as well as the development of hardware that implements geometric calculation methods and provides a new graphical interface. The concept developed by the authors is aimed at creating specialized accelerators of geometric transformations.

Potential strategies to mitigate the heat island impacts of highway pavement on megacities with considerations of energy uses

With the rapid expansion of urbanization around the world, the urban heat island effect embraced an enormous negative impact on megacities, including energy, environment and health conditions. Unfortunately, constructive geometry and human activities of megacities severely intensified the urban heat island phenomenon. Giving the contributions of pavement on the deterioration of urban heat island in megacities, this literature review synthesized the state-of-the-art potential strategies of pavement cooling technologies and applications to mitigate the urban heat island effect. Firstly, the optical-thermal balance on surface and the interior thermal behavior of pavement were comprehensively analyzed. Afterwards, this paper redefined the cool pavement technologies as the passive mitigation strategies, involved the application of reflective pavement, evaporation pavement, heat-induced pavement and phase change materials. Especially, urban canyon impact and urban design were emphasized, which tended to be neglected on the implementations of cool pavement. In addition, the solar collector technologies on the pavement were proposed to sustainably reuse solar energy, acting as active strategies and renewable sources. Typically, representatives of solar collectors, including heat collection and thermoelectric pavement, were described and investigated. Existing research results presented the promising potential of pavement cooling technologies to mitigate the urban heat island impact and renewable energy sustainability. Moreover, prospects were also summarized to show the opportunities and avenues for future study.

A Methodology for Buildings Access to Solar Radiation in Sustainable Cities

The growing need to improve the environmental and energy sustainability of buildings involves the use of solar radiation incident on their surfaces. However, in cities, this task is complicated due to the constructive geometry that leads to shading between buildings. In this context, this work presents a study of solar access to the façades of buildings in cities. The methodology is based on the determination of the incident annual solar radiation in 121 significant points of each façade considering the twelve representative days of the year. To characterize the influence of the different city typologies on solar access, the urban solar coefficient is proposed. A study of two neighborhoods in Cordoba (Spain) with different urban settings have been analyzed. Specifically, two typologies of neighborhoods have been compared: one with “L-shaped” and “U-shaped blocks” and another with “Grouped blocks”. For both of them, the Urban Solar Coefficient has been calculated, obtaining a higher mean value for the neighborhood with “L-shaped” and “U-shaped blocks” (0.317) than for the one with “Grouped blocks” (0.260). Accordingly, the results show that urban morphology can influence the Urban Solar Coefficient and solar access. Finally, a regression model for each neighborhood has been obtained in order to determine the dependence of the Urban Solar Coefficient on neighborhood geometry factors.

Приближенное решение задачи о квадратуре круга

It is known that squaring the circle (the problem consisting in construction of a square with the same area as a given circle), together with duplication of cube and angle trisection, is one of the most famous unsolv able problems of constructive geometry for construction with compass and straightedge. The solution of squaring the circle problem is reduced to the straightening of the circle, that is, to the construction of a segment equal in length to the circle, and its insolvability is connected with the pi character transcendence. In this paper, the limiting case of one of Christian Huygens theorems, which establishes an estimate for length of circumference of a circle through perimeters of regular polygons inscribed in circle and circumscribed about it, is proved. On this basis has been proposed and justified an approximate method for squaring the circle problem solving, which allows consistently construct arbitrarily exact solutions of the problem. We will approximate an arc of a circle whose radius is a multiple of the given circle’s radius, with the help of a segment which is parallel to a shrinking it chord, and then will increase or decrease this segment in the required number of times, so that the resulting segment’s length would be approximately equal to half of the given circle’s circumference. The approximation accuracy will be the higher the smaller arc of the circle we will consider. But possibilities of real tools are limited, and not suitable for both too small and too large drawing scales. In order to cope with this problem, an algorithm for scaled approximation has been proposed, in which it is sufficient to increase (or reduce) the drawing fragment, so that all the time sta y within the sheet of the same size. Perhaps this approach will be useful for other constructions, including the exact ones, where it is necessary to come to very large or vice versa very small drawings’ dimensions.

The Analytical Approach to the Parameterized Fuzzy Operators Design

In this paper we discussed the problem of generation and efficient software- and hardware-based implementation of branches of fuzzy logic parametric operations that can be used as basic operations of reconfigurable fuzzy systems (embedded fuzzy controllers). The ways of design of strict t-norms and t-conorms on fuzzy systems in presented paper based on the mathematical theory of geometrical aggregation operations from the Rvatchev constructive geometry. Presented theoretical results may be implemented for a new branches of membership functions design and in the hardware-based units of analytical fuzzy inference systems.

Geometric problem solving with strings and pins

ABSTRACTHumans solve spatial and abstract problems more easily if these can be visualized and/or physically manipulated. We analyze the domain of geometric problem solving from a cognitive perspective and identify several levels of domain abstraction that interact in the problem solving process. We discuss the roles of physical manifestations of spatial configurations, their manipulation, and their perception for understanding problem solving processes. We propose an extension of the classical problem solving repertoire of constructive geometry to approach certain problems more directly than under the compass-and-straightedge paradigm. Specifically, we introduce strings and pins as helpful metaphors for a generalization of the constructive geometry approach. We present classical problems from spatial problem solving to illustrate the ‘strings and pins’ paradigm. Three case studies are discussed: strings-and-pins solutions to (i) the ellipse construction problem; (ii) the shortest path problem; and (iii) the angle trisection problem. Comparisons to formal solutions are drawn. Differences and similarities between the compass-and-straightedge paradigm and the strings-and-pins paradigm are analyzed. Features and limitations of constructive and depictive geometry as well as implications for computational approaches are discussed. The strings-and-pins domain is shown to be more general and less restrictive than the compass-and-straightedge domain.

Formalizing Constructive Projective Geometry in Agda

We present a formalization of Projective Geometry in the proof assistant and programming language Agda. We formalize a recent development in constructive Projective Geometry which has been shown appropriate to cover most traditional topics in the area applying only constructively valid methods. The equivalence with other well-known constructive axiomatic systems for projective geometry is proved and formalized.The implementation covers a basic fragment of intuitionistic synthetic Projective Plane Geometry including the axioms, principle of duality, Fano and Desargues properties and harmonic conjugates.We focus in an illustrative example of implementation of a complex and large proof which appears partially and vaguely described in the literature; namely the uniqueness of the harmonic conjugate.The most important details of our implementation are described and we show how to take advantage of several interesting properties of Agda such as modules, dependent record types, implicit arguments and instances which result very helpful to reduce the typical verbosity of formal proofs.

SCAFFOLDS AND DISSECTIONS: COMPUTATIONAL RECONSTRUCTION OF INDIC TEMPLES AND THEIR ARCHITECTURAL PRODUCTION

This paper presents scaffolds and dissections for the comparative representation and analysis of Indic temple geometry found in temples across South and Southeast Asia. Scaffolds are representational schema that capture classes of constructive geometry such as grids, geometric profiles and procedures from canonical descriptions. Dissections represent horizontal and vertical profiles that capture attribute features of a temple from field measurement methods. Together, they guide the development of 3D schematic reconstructions of individual temples that capture the architectural form of the temple, but also the knowledge of temple production and their architectural lineage. Drawing upon canonical descriptions and previous scholarship on temple geometry, the paper outlines schematic reconstructions of four individual temples. A comparative analysis of the similarities and differences between the temples reveals the role of the canonical constructive mechanisms underlying these temples. Individual temples and their similarity features are categorised into a series of related objects in compliance classes. More broadly, both method and vehicle posit a broader understanding of how individual buildings of a particular historical and philosophical lineage may be compositionally connected through the method of scaffolds and dissections and provide a symbolic view of variance in architectural production over time.

Визуально-графическое проектирование единой конструктивной модели для решения аналогов задачи Аполлония с учетом мнимых геометрических образов

The Apollonius problem on construction of circles, tangent to three arbitrary given circles of a plane, is one of classical geometry’s well-studied problems. The presented paper’s materials are directed at development a unified theory for Apollonius problem solving, taking into account it’s not only real, but also invisible complex-valued images. In the paper it has been demonstrated, that fundamental geometric structures, on which Apollonius problem is based on, are applicable not only to real, but also to complex-valued data, that makes possible to eliminate many exceptions, currently existing in it. In this paper Apollonius problem’s fundamental nature and its strong correlation with projective and quadratic geometric transformations has been disclosed. It has been proved that Apollonius problem and its analogues have a single solution method, in contrast to the prevailing idea that these problems can be solved only by separate particular methods. A concept of geometric experiment proposed by the author has allowed find out many previously unknown and discussed in this paper common factors, due to the set of many computational tests in the system Simplex for visual design of geometric models. In this paper is considered an example for solving an analogue of Apollonian problem for three-dimensional space, but proposed algorithm’s operation is universal, and it can be equally applied to solving similar problems in spaces of arbitrary dimensions. Obtained results demonstrate capabilities of methods for constructive modeling and multidimensional descriptive geometry in application to solving of complex mathematical problems, and determine the trends in development for automation systems of constructive geometric modeling.

ICVG : Practical Constructive Volume Geometry for Indirect Visualization

ICVG : Practical Constructive Volume Geometry for Indirect Visualization

Rationalisation of the geometry for production of the 80-m glazed Sphere for the Expo 2017 in Astana

The emblematic entrance pavilion for the Expo 2017 in Astana is the so-called ‘Sphere’, which with its 80-m diameter is worldwide the largest closed sphere completely covered with glass. The continuous external surface of the Sphere was designed to be built with spherically curved glass panels of parallelogramic shape. The panels are fixed at their perimeter to a structural grid of bent round tubes of steel. This secondary grid is fixed in turn to a coarser primary structural grid at evenly spaced supports. The purpose of this article is to give insight into the constructive geometry design of the Sphere. One of the main targets of ‘geometric design for production’ is to specify the geometry of a construction in such a way that fabrication and installation procedures can be simplified and carried out with a minimum of effort, cost and time. The main measure for the Sphere was to simplify the basic grid by replacing all original three-dimensional free-form curves with series of interconnected circular arcs. Thus, a typical curved tubular segment could be bent on the plane with a constant radius. A further approach to get a simple and sound construction eliminating geometric torsion and minimising eccentricities among construction subsystems, for example, glazing, sealing, fixings and substructure, was the extensive usage of conical systems inherent in the geometry of the sphere for orienting and positioning the various subsystems components. These two geometric simplification measures were determinant for the practical and economic implementation of the construction, fabrication and installation of the Sphere.