Velaroidal Shells and Shells of the Velaroidal Type

Abstract. A large number of new surfaces are presented, formed by congruent curves with variable curvature, but remaining in the same class, and superellipses. All surfaces are included in the class-es Rotation Surfaces, Velaroidal Translaation Surfaces, and Algebraic Surfaces with a Carcass of Three Main Flat Curves. All surfaces of the same class are defined by the same general explicit and para-metric equations, and thanks to the presence of many constants in the superellipse equation, it is possible to obtain a lot of known and new surfaces. Despite the fact that the method of construction of the considered surfaces is known, in the presented article it is il-lustrated and visualized on many examples. The surfaces were constructed using a numeric computing environment MATLAB. The surfaces of a general-view superellipse were built on the basis of a new computer program that allows them to be visualized in a multimedia mode by a set change in the exponents contained in the meridian-superellipse formula. All built rotation surfaces have a common name — superellipsoids of rotation. For the first time it is shown that algebraic surfaces with a given frame in three mutually perpendicular planes, applied in shipbuilding, can also find application in the architecture of public buildings. Superellipses are used as the rigid frame of surfaces. In the overview section of the article on the basis of the available publications it is shown that the geometry of the form affects the stress-deformable state of shells with the proposed medial surfaces. The materials of the article give an opportunity in the future to find the optimal shells outlined on the considered analytical surfaces of three different classes, which are considered in the article, taking into account the criteria of optimality applied in architecture, construction, engineering and shipbuilding.

Computer Simulation of New Forms of Shell Structures

Strashnov

2023

Synthetic Representation of the "Oblique Symmetry" Transformation Using the Example of an Ellipse

Rustamyan,

Bayanov,

Slavin

2023

Geometric transformations play a pivotal role in computer graphics, determining the position and shape of objects. In machine learning, they are applied for processing and analyzing data, such as in images. In geometric surface modeling, they are utilized for the creation and transformation of three-dimensional forms. In physics, geometric transformations assist in describing the motion of objects in space and time. The aim of this work is to analyse and study the geometric transformation known as "oblique symmetry." Primarily, the article seeks to elucidate a number of important properties of this transformation, expanding the field of knowledge in perspective-affine correspondence. Throughout the study, the principal directions of oblique symmetry are identified, and their relationship with the axis and direction of the transformation is established. It is crucial to emphasise that the analysis makes it evident that the axis and the direction of symmetry are equivalent and interchangeable. Additionally, the article addresses the challenge of transforming an arbitrary ellipse, defined by its semi-axes, into a circle of equal area. In this context, a method is proposed to determine the axis and direction of oblique symmetry for a given ellipse. Based on the results obtained and the analysis conducted, the authors propose a geometric algorithm that provides the capacity to resolve positional problems in the field of descriptive geometry. This algorithm also offers a novel method for constructing ellipses with given semi-axes, which holds practical significance in various engineering and geometric issues. In the conclusion of the article, a specific example of applying the developed method is provided, clearly demonstrating its practical value and real capabilities in solving positional problems in the field of descriptive geometry. Moreover, directions for future research in the field of shape formation are suggested, utilising the "oblique symmetry" transformation in the spaces and .

Building the Perspective of a Straight Line Along The Picture Trail and the Vanishing Point

Sal'kov

2024

The article recalls the essence of academic subjects: descriptive geometry, drawing, engineering graphics, computer graphics and what each of these subjects is intended for. It is said about the compulsory study of geometry by students, especially those students who study in technical fields of study, about the role of descriptive geometry in the learning process and later in practical activities. Therefore, the existence of departments of geometric and graphic profile for each technical university is unconditionally necessary. It is explained that the modern disdainful attitude of the university leadership towards the departments of the geometric direction should not lead to the disappearance of these departments, since an engineer without knowledge of geometry is a fundamental underachiever, and at best an amateur. The reasons that led to the insufficiency of highly qualified specialists in the geometric and graphic profile in universities are given and, in this regard, some steps are proposed to improve the quality of engineering education. The situation of the educational process in the supposedly leading technical universities of the country, which have already got rid of the departments of the geometric direction or are on the road to this "great" achievement, is given as a negative example, as a result of which various information has been received for quite a long time about the not very favorable situation in these universities. Some measures are proposed to improve geometric education in Russian universities.