Properties of Dupin Cyclide and Their Application. Part 4: Applications

2016 ◽  
Vol 4 (1) ◽  
pp. 21-33 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Dense Packing ◽  
Descriptive Geometry ◽  
Practical Application ◽  
Surfaces Of Revolution ◽  
Special Cases ◽  
Dupin Cyclide ◽  
Different Diameters

In the first and second parts of the work there were considered mainly properties of Dupin cyclide, and given some examples of their application: three ways of solving the problem of Apollonius using only compass and ruler, using the identified properties of cyclide; it is determined that the focal surfaces of Dupin cyclid are degenerated in the lines and represent curves of the second order – herefrom Dupin cyclide can be defined by conic curve and a sphere whose center lies on the focal curve. Polyconic compliance of these focal curves is identified. The formation of the surface of the fourth order on the basis of defocusing curves of the second order is shown. In this issue of the journal the reader is invited to consider the practical application of Dupin cyclide’s properties. The proposed solution of Fermat’s classical task about the touch of the four spheres by the fifth with a ruler and compass, i.e., in the classical way. This task is the basis for the problem of dense packing. In the following there is an application of Dupin cyclide as a transition pipe element, providing smooth coupling of pipes of different diameters in places of their connections. Then the author provides the examples of Dupin cyclide’s application in the architecture as a shell coating. It is shown how to produce membranes from the same cyclide’s modules, from different modules of the same cyclide, from the modules of different cyclides, from cyclides with the inclusion of other surfaces, special cases of cyclides in the educational process. The practical application of the last problem found the place in descriptive geometry at the final geometrical education of architects in the "Construction of surfaces". Here such special cased of cyclides as conical and cylindrical surfaces of revolution.

Properties of Dupin Cyclide and Their Application. Part 3: The Problem of Apollonius

2015 ◽  
Vol 3 (4) ◽  
pp. 3-14 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Practical Application ◽  
Practical Utility ◽  
Engineering Graphics ◽  
The Third ◽  
Focal Surface ◽  
Dupin Cyclide ◽  
Straight Lines

In the first part of work was addressed mainly the issue of properties under Dupin cyclide, and given some examples of their applications: three ways of solving the problem of Apollonius using only compass and ruler, using the identified properties of Dupin cyclid. The second part of work continued with consideration of the use of property under a lie of Dupin. It is determined that the focal surface of cyclid of Dupin is degenerated in the lines and represent curves of the second order. Here under a lie can be defined conic curve and a sphere whose center lies on the focal curve. Polyconic conformity these focal curves is revealed. The article show the formation of the surface of the fourth order on the basis of defocusing curves of the second order. In this issue of the journal the reader is invited to consider the practical application of properties under a lie of Dupin for example of well known problems with on-voltage. If the first part of the work was cited only three ways of solving the problem of Apollonius, in the third part the author considers other possible mates: as at zero the size of the radius of the circle and the demon is of course great. All decisions – both known and not really based on properties of Dupin cyclide. In the course of engineering graphics, introductory tests, as they say now, drawing on architectural faculties there are tasks, dedicated to the mating arcs of circles with straight lines, and circles passing through the points in various combinations. Therefore, the proposed practical application cannot be considered far-fetched – it is based on the practical utility of method.

Properties of Cyclide Dyupen and Their Application. Part 1

2015 ◽  
Vol 3 (1) ◽  
pp. 16-25 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Nineteenth Century ◽  
Contact Line ◽  
Classical Problem ◽  
Descriptive Geometry ◽  
Practical Application ◽  
Surfaces Of Revolution ◽  
Early Nineteenth Century ◽  
Focal Surface ◽  
Lines Of Curvature ◽  
And Task

In the training course in descriptive geometry we consider the class of surfaces formed by circles and named "Circular surface. Within this class of surfaces is the so-called kanalowe surface. Under a lie cyclide belong to canalave surfaces, but in the course of descriptive geometry, their formation is not considered. Under a lie cyclide were discovered by Pierre Charles Francois Dyupen in the early nineteenth century and named in his honor. He dyupen was a disciple of Gaspard Monge, like many great scientists in France at that time. Under a lie cyclide usually represented as envelopes of a family of spheres tangent to three given. Under a lie – the only surface whose focal surface degenerates into a line, and all lines of curvature are circles. Particular cases of ticlid cyclide is a torus, and conical and cylindrical surfaces of revolution. The paper discusses the analytical representation of the focal lines for the General case of a job under a lie cyclide. It is analytically proved that the contact line inscribed in cyclide spheres are circles, and degenerate in the focal curve on the surface is a curve of the В учебном курсе начертательной геометрии из- учается класс поверхностей, образованный окруж- ностями и названный «Циклические поверхности» [5; 8; 12]. Внутри этого класса поверхностей есть так называемые каналовые поверхности. Циклиды Дюпена принадлежат к каналовым поверхностям, более того, они являются частным случаем [2–4; 6] этих поверхностей, но в курсе начертательной гео- метрии их формирование не рассматривается. Циклиды Дюпена были открыты Пьером Шарлем Франсуа Дюпеном (1784–1873) в начале XIX в. и названы в его честь [14]. Дюпен (рис. 1) был учени- ком Гаспара Монжа, как и многие великие ученые Франции того времени, и являлся почетным членом Петербургской академии наук c 20 декабря 1826 г. second order. Identified some (nine) properties of this surface. As a practical application of ticlid cyclide solved such well-known classical problem as the problem of Apollonius (about Casa-NII three circles fourth) and task Farm (touch four spheres fifth) using again the classic way – with a ruler and a compass. In the first part of the article is only three ways to solve the problem of Apollonius solely by means of compass and ruler, using the properties of cyclide Dyupen.

General Analysis of the Shape of Two Similar Second-Order Surfaces’ Intersection Line

2021 ◽  
pp. 24-34
Author(s):  
A. Ivaschenko ◽  
D. Vavanov
Keyword(s):  
Fourth Order ◽  
Relative Position ◽  
Second Order ◽  
Point Of View ◽  
General Analysis ◽  
Surface Shape ◽  
Descriptive Geometry ◽  
Shape Parameters ◽  
Analytical Geometry ◽  
Interacting Surfaces

The presented paper is devoted to classification questions of fourth-order spatial curves, obtained as a result of intersection of non-degenerate second-order surfaces (quadrics) from the point of view of the forms of the original quadrics generating this curve. At the beginning of the paper is performed a brief historical overview of appearance of well-known and widely used curves ranging from ancient times and ending with the current state in the theory of curves and surfaces. Then a general analysis of the influence of the shape parameters and the relative position of original surfaces on the shape of the resulting curve and some of its parameters (number of components, presence of singular points, curve components flatness or spatiality) is carried out. Curves obtained as a result of intersection of equitype surfaces are described in more detail. The concept of interacting surfaces is introduced, various possible cases of the forms of the quadrics generating the curve are analyzed. A classification of fourth-order curves based on the shape parameters and relative position of second-order surfaces is proposed as an option. Illustrations of the resulting curve shapes with different shape parameters and location of generating quadrics are given. All surfaces and curves are considered in real affine space, taking into account the possibility of constructing them using descriptive geometry methods. Possible further research directions related to the analysis of the curves under discussion are briefly considered. In addition, are expressed hypotheses related to these curves use in the process of studying by students of technical universities the courses in analytical geometry, descriptive geometry, differential geometry and computer graphics. The main attention is paid to forms, therefore a wide variability of the surface shape in the framework of its described equation has been shown, provided by various values of numerical parameters.

Dupin Cyclide and Second-Order Curves. Part 1

2016 ◽  
Vol 4 (2) ◽  
pp. 19-28 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Second Order ◽  
Descriptive Geometry ◽  
Analytical Equation ◽  
Not Given ◽  
Academic Literature ◽  
Order Curve ◽  
Plane Passing ◽  
Dupin Cyclide ◽  
The One ◽  
Order Smoothness

Making smooth shapes of various products is caused by the following requirements: aerodynamic, structural, aesthetic, etc. That’s why the review of the topic of second-order curves is included in many textbooks on descriptive geometry and engineering graphics. These curves can be used as a transition from the one line to another as the first and second order smoothness. Unfortunately, in modern textbooks on engineering graphics the building of Konik is not given. Despite the fact that all the second-order curves are banded by a single analytical equation, geometrically they unites by the affiliation of the quadric, projective unites by the commonality of their construction, in the academic literature for each of these curves is offered its own individual plot. Considering the patterns associated with Dupin cyclide, you can pay attention to the following peculiarity: the center of the sphere that is in contact circumferentially with Dupin cyclide, by changing the radius of the sphere moves along the second-order curve. The circle of contact of the sphere with Dupin cyclide is always located in a plane passing through one of the two axes, and each of these planes intersects cyclide by two circles. This property formed the basis of the graphical constructions that are common to all second-order curves. In addition, considered building has a connection with such transformation as the dilation or the central similarity. This article considers the methods of constructing of second-order curves, which are the lines of centers tangent of the spheres, applies a systematic approach.

Representations of Dupin Cyclides

2017 ◽  
Vol 5 (3) ◽  
pp. 11-24 ◽  
Author(s):  
Николай Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Second Order ◽  
Single Parameter ◽  
Geometric Constructions ◽  
Order Curve ◽  
Computer Technologies ◽  
Special Cases ◽  
Dupin Cyclide ◽  
Dupin Cyclides ◽  
Confocal Conics

We know very little about such an interesting surface as Dupin cyclide. It belongs to channel surfaces, its special cases are tor, conical and cylindrical surfaces of rotation. It is known that Dupin cyclides are the only surfaces whose focal surfaces, that are surfaces consisting of sets of curvatures centers points, have been degenerated in second-order curves. Two sets give two confocal conics. That is why any study of Dupin cyclides is of great interest both scientific and applied. In the works devoted to Dupin cyclide and published in the "Geometry and Graphics" journal, are presented various properties of cyclides, and demonstrated application of these surfaces in various industries, mostly in construction. Based on the cyclides’ properties in 1980s have been developed numerous inventions relating to devices for drawing and having the opportunity to be applied in various geometric constructions with the use of computer technologies. In the present paper have been considered various options for representation of Dupin cyclides on a different basis – from the traditional way using the three given spheres unto the second-order curves. In such a case, if it is possible to represent four cyclides by three spheres, and when cyclide is represented by the second-order curve (konic) and the sphere their number is reduced to two, then in representation of cyclide by the conic and one of two cyclide’s axes a single Dupin cyclide is obtained. The conic itself without any additional parameters represents the single-parameter set of cyclides. Representations of Dupin cyclides by ellipse, hyperbola and parabola have been considered. The work has been sufficiently illustrated.

scholarly journals Integrity basis for a second-order and a fourth-order tensor

1982 ◽  
Vol 5 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Tensor Function ◽  
Order Tensor ◽  
Isotropic Tensor ◽  
Anisotropic Properties ◽  
Isotropic Tensor Function ◽  
Fourth Order Tensor

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

scholarly journals Simplified and improved criteria for oscillation of delay differential equations of fourth order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. Moaaz ◽  
A. Muhib ◽  
D. Baleanu ◽  
W. Alharbi ◽  
E. E. Mahmoud
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Interesting Point ◽  
All Solutions ◽  
Special Cases ◽  
Delay Differential ◽  
Noncanonical Operator

AbstractAn interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

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