Surfaces of congruent sections on cylinders

Introduction. The definition of surfaces of congruent sections was first formulated in the work written by I.I. Kotov. These and several other types of surfaces, generated by the motion of a curve, belonged to the class of kinematic surfaces. Such kinematic surfaces as those of plane parallel displacement, surfaces of rotation, Monge surfaces, cyclic surfaces with ge-nerating circles having constant radius, rotative and spiroidal surfaces, helical some helix-shaped surfaces can be included into the class of surfaces that have congruent sections. Materials and methods. Using I.I. Kotov’s methodology, the authors first derived parametrical and vector equations for eight surfaces of congruent pendulum type cross sections of circular, elliptic, and parabolic cylinders and several helix-shaped surfaces. Circles, ellipses, and parabolas, located in the plane of the generating curve of a guiding cylinder or in the planes of a bundle that passes through the longitudinal axis of a cylinder, generate plane curves. Ellipses, analyzed in the article, can be easily converted into circles and this procedure can increase the number of shapes analyzed here. Results. Formulas are provided in the generalized form, so the shape of a plane generating curve can be arbitrary. Some surfaces of congruent sections are determined by two varieties of parametric equations. In one case, the central angle of the guiding cylindrical surface was used as an independent parameter, but in the other case, one of rectangular coordinates of the cylinder’s guiding curve served as an independent parameter. Two types of surfaces are analyzed: 1) when local axes of generating curves remain parallel in motion; 2) when these axes rotate. Conclusions. The analysis of the sources and the results, recommendations and proposals for application of surfaces, having congruent sections, is made with a view to their use in architecture and technology. The list of references has 27 positions, and it shows that the surfaces considered in this paper are being analyzed by architects, engineers, and geometricians both in Russia and abroad.

90-Year Anniversary of Mitkht’s Engineering Graphics Chair

In May 2018 the Engineering Graphics Chair celebrates 90 years from the date of its foundation. The Chair was organized in 1928. The paper tells the Chair’s history, its teachers and heads, as well as a brief description of its scientific work. In 1900 were established the Moscow Higher Feminine Courses (MHFCs). A year after the October revolution, in late 1918, MHFCs were transformed into the 2nd Moscow state University. In 1930 the 2nd MSU was reorganized into three independent institutes: medical, chemical-technological and pedagogical ones. In May 1928 was organized the Chair of Technical Drawing, this moment is the counting of Engineering Graphics Chair existence. The first head of the Chair was S.G. Borisov. Than the Chair was supervised by Associate Professor A.A. Sintsov (from September 1932 till January 1942), Associate Professor M.Ya. Khanyutin (in 1942–1952), Associate Professor N.I. Noskov (in 1954–1962), Associate Professor F.T. Karpechenko (in 1962–1972), Senior Lecturer N.A. Sevruk (in 1972–1982), Professor, Doctor of Engineering E.K. Voloshin-Chelpan (from January 1982 to August 2007), Associate Professor V.I. Vyshnepolsky (from August 2007 till present). Currently, on the Chair are carrying out researches in the following directions: Higher School’s Pedagogy; Academic Competitions of Regional and All-Russia’s Level; Loci; Geometry of Cyclic Surfaces; Theory of Kinetic Geometry; Geometries; Geometric Transformations; Theory of Fractals; Famous Geometers’ Biographies.

Formation of Surfaces Under Kinetic Displaying

This work is the development of previously published ones in the journal "Geometry and Graphics" as follows: "Kinematic Correspondence of Rotating Spaces" (№ 1, 2013) and "Formation of Cyclic Surfaces in Kinetic Geometry" (№ 4, 2017). Many of mechanisms make rotational movement, wherein rotating parts of one mechanism "invade" into the zone of rotation for another rotating mechanism’s parts. At the same time, in addition to rotation, they can make other movements, both translational and rotational nature. The theory of kinetic geometry, of which this work is an integral part, is developed in order to avoid collisions of two or more parts of different mechanisms with each other. This is a rather complicated problem in mechanical engineering, in the mining industry, in metallurgy, and in space navigation, where there are no objects that are at rest. Therefore, the kinetic theory of matching for rotating spaces R1 3 and R23 when they are independent from each other movement is quite relevant. In this work have been considered cases for mapping of geometric figures of one space to another one when these figures are moving inside their space R13 . A theory which is presented has been called kinetic geometry, as it relates to engineering problems associated with gearings. These problems were addressed for the first time and drew-up as inventions. A monograph entitled "Introduction to Kinetic Geometry" is currently being prepared for publication.

Formation of Cyclic Surfaces in Kinetic Geometry

This paper is an evolution of the "Kinematic Compliance of Rotating Spaces" paper, previously published in the "Geometry and Graphics" journal №1, 2013. A great many of mechanisms are making rotational movement, wherein rotating parts of one mechanism are "invading" into a rotation zone belonging to parts of another rotating mechanism. The challenge is to prevent the collision of rotating parts belonging to two or more details with each other. This problem is particularly sensitive for machine engineering. In space navigation, where, in principle, there are no objects that are at rest, the problem of satellites collision with astronomical bodies rotating around their axes is also the urgent one. Therefore, the theory of kinematic matching for rotating spaces R31 and R32 when they are moving independently from each other is urgent too. Each of two considered spaces may have a uniform or non-uniform movement in a given direction, a curved movement or a rotational movement around the axis specified for each space. In this paper has been considered the formation of cyclic surfaces obtained by rotation of one space relative to another one and different orientations of the generating line relative to the axes. Has been considered one of the options for rotating spaces, when their axes are parallel. In such a case the generating line is located in the following positions: it is straight and parallel to the axis; it is straight and intersects the axis; the rectilinear generator is in a plane that is parallel to the plane of the axes; the generating line is a straight line of general position; the generating line is a space curve. Has been demonstrated application of the rotating spaces theory in mining, chemical and machine tool industries, made in the form of inventions, confirmed by copyright certificates of the USSR.

On Errors in Terminology on Theory of Surfaces And Geometric Modelling

At present, a great amount of scientific papers, monographs, and reference books dealing with analytical and differential geometry of surfaces have been published. They contain materials for following geometric investigations, for implementation of received earlier geometrical results into architecture, building, and machinery manufacturing. In the paper it has been shown on the specific examples that sometimes the results of geometric investigations for shells’ middle surfaces taken in published references for the following application without check could lead to serious errors because of ones in the surfaces equations or inexactitudes in a surfaces class definition. At present, 38 classes of surfaces, uniting more than 600 ones that have their own names and are described in scientific publications, are known. The author has worked up a great number of researches and found errors, inaccuracies, and alternative versions in monographs and scientific papers, related to questions on geometry of developable surfaces (conic and torse surfaces), surfaces of rev olution (paraboloid and ellipsoid of revolution, nodoid), minimal surfaces (catenoids), conoids, and cyclic surfaces including the canal ones. In actual practice there are much more geometric errors, but in this paper are discussed only well-known geometricians and architects’ works, as well as in this paper there is no information on surfaces that are presented at specialized sites in Internet. Here are encountered misreckoned coefficients for surfaces’ fundamental quadratic forms, there are errors in the formulae for the quadratic forms’ coefficients determination, as well as in the formulae for the calculation a surface element’s area, surface’s principle curvatures, and so on. All of encountered errors have been divided into four groups. The fourth group’s errors named as “typographical errors and authors’ slips of the pen” have been considered fragmentarily because they are encountered the most frequently, and can be corrected by the authors themselves in the following papers.