Competency approach to the study of the second-order surface interface as the geometric location of points in space

The necessity of the formation of students' functional literacy as a competency approach to the training of future Mathematics teachers is substantiated on the example of studying of one of the topics of analytical geometry. It has been established that a prerequisite for the development of any competency prescribed in the standards of secondary education is the initial existence of a sufficiently new concept of functional literacy for a student of a certain level. The basic literacy comes down to the ability to read, write and express of one's thoughts correctly. Let us consider the issue of functional literacy from the point of view of the pedagogic specialty. Acquaintance with the well-known textbooks of analytic geometry allows us to say that 2nd order algebraic surfaces in Euclidean space are determined in most cases algebraically by means of equations. A constructive approach is also of use – surfaces are obtained by rotating 2nd degree curves around their symmetry axes and by deformation of the resulting surfaces by compression. The metric approach, as it used for 2nd order curves, is restricted only by the formulation of problems to find the certain locus of points in space. The exception is the article Dmitriy Perepyolkin which was published in 1936. In this paper the locus of points in space with the following characteristic property is studied – the ratio of the distance to a given point to the distance to a given straight line is constant. The strait line is assumed not to contain the point. The study is held out in pure geometrical manner – it is done using the method of sections and known loci of points on the surface. In the present article we study the locus of points in space defined by metric relation to a certain set of pairs of points, lines and planes. It is shown that any non-degenerate 2nd order surface can be considered as a certain locus of points of space and this interpretation is not unique.

Offset Ruled Surface in Euclidean Space with Density

In this paper, offset ruled surfaces in these spaces are defined by using the geometry of ruled surfaces in Euclidean space with density. The mean curvature and Gaussian curvature of these surfaces are studied. In addition, the relationships between the mean curvature and mean curvature with density, and the Gaussian curvature and the Gaussian curvature with density of the offset ruled surfaces in E 3 with density e z and e − x 2− y 2 are given.

Construction of constant mean curvature n-noids using the DPW method

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

Osculating surfaces of curves on surfaces

Oosculating sphere have been studied in classical differential geometry [1]. In this article the osculating surfaces of higher order of space curves on surfaces in Euclidean space is considered. We study the intrinsic differential geometry of curves on surfaces by analyzing their contact with surfaces of higher order.

Skew mean curvature flow

The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space [Formula: see text]. A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest.