Determining regularities in the construction of curves and surfaces using the Darboux trihedron

Authors

DOI:

https://doi.org/10.15587/1729-4061.2023.279007

Keywords:

accompanying trihedron, Darboux trihedron, parametric equations of curves and surfaces, transition formulas

Abstract

The Frénet trihedron, known in differential geometry, is accompanying for a spatial and, as a special case, for a flat curve. Its three mutually perpendicular unit orts are defined uniquely for any point on the curve except for some special ones. Unlike the Frénet trihedron, the Darboux trihedron relates to the surface. Two of its unit orts are located in a plane tangent to the surface, and the third is directed normally to the surface. It can also be accompanying for the curve, which is located on the surface. To this end, one of the orts in the plane tangent to the surface must be tangent to the curve.

Trihedra are movable and, with respect to a fixed coordinate system, change their position due to movement and rotation. The object of research is the process of formation of curves and surfaces, as a result of the geometric sum of the bulk motion of the Darboux trihedron and the relative motion of the point in its system under given conditions. In the study of the geometric characteristics of curves and surfaces, it is necessary to have formulas for the transition from the position of the elements of these objects in the system of a moving trihedron to the position in a fixed Cartesian coordinate system. This is exactly what needs to be solved. The results obtained are parametric equations of curves and surfaces that are tied to the initial surface. Nine guide cosines were found, three for each ort.

A distinctive feature of this approach in comparison with the traditional one is the use of two systems: fixed and mobile, which is the Darboux trihedron. This approach allows us to consider in a new way the problem of the construction of curves and surfaces. The scope of practical application can be the construction of geometric shapes on a given surface. An example of such a construction is the laying of a pipeline along a given line on the surface. In addition, the sum of the relative motion of a point in a trihedron and the bulk motion of the trihedron itself over the surface gives an absolute trajectory of motion. Its sequential differentiation produces absolute speed and absolute acceleration without finding individual components, including the Coriolis acceleration. This could be used in point dynamics problems

Author Biographies

Ali Kadhim Ahmed, University of Diyala

PhD, Lecturer

Department of Soil Science and Water Resources

College of Agriculture

Andrii Nesvidomin, National University of Life and Environmental Sciences of Ukraine

PhD, Associate Professor

Department of Descriptive Geometry, Computer Graphics and Design

Serhii Pylypaka, National University of Life and Environmental Sciences of Ukraine

Doctor of Technical Sciences, Professor, Head of Department

Department of Descriptive Geometry, Computer Graphics and Design

Tatiana Volina, National University of Life and Environmental Sciences of Ukraine

PhD, Associate Professor

Department of Descriptive Geometry, Computer Graphics and Design

Serhii Dieniezhnikov, Sumy State Pedagogical University named after A.S. Makarenko

PhD, Associate Professor

Department of Management of Education and Pedagogy of the Higher School

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Determining regularities in the construction of curves and surfaces using the Darboux trihedron

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Published

2023-06-30

How to Cite

Ahmed, A. K., Nesvidomin, A., Pylypaka, S., Volina, T., & Dieniezhnikov, S. (2023). Determining regularities in the construction of curves and surfaces using the Darboux trihedron. Eastern-European Journal of Enterprise Technologies, 3(1 (123), 6–12. https://doi.org/10.15587/1729-4061.2023.279007

Issue

Vol. 3 No. 1 (123) (2023): Engineering technological systems

Section

Engineering technological systems

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Copyright (c) 2023 Ali Kadhim Ahmed, Andrii Nesvidomin, Serhii Pylypaka, Tatiana Volina, Serhii Dieniezhnikov

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