Devising a technique for constructing tubular surfaces referred to a coordinate grid of lines of curvature
DOI:
https://doi.org/10.15587/1729-4061.2026.354601Keywords:
curvature lines, slope curve, arc length, Frenet trihedron, orthogonal gridAbstract
This study considers the construction of tubular surfaces with a spatial axis of slope, referred to a coordinate grid of curvature lines. Such surfaces have a number of mathematical advantages compared to surfaces described by arbitrary coordinate grids. In differential geometry, this has a theoretical justification and applied value. This follows from the special role of curvature lines as geometrically privileged directions on a surface with minimal and maximal curvatures.
To parameterize a tubular surface in this way, it is necessary that the length of its axis be described by analytical dependences in a finite form. Typically, the length of spatial curves is determined by numerical integration. There is a known group of plane curves that are described by parametric equations as a function of the arc length and for which such a problem does not exist. This work proposes taking such curves as a horizontal projection of a spatial curve. The spatial curve should be constructed as a slope curve with a constant elevation angle relative to the horizontal plane. Then the spatial curve, the equations of which include the elevation angle, will be described as a function of the arc length. Its use as the axis of the tubular surface makes it possible to attribute the latter to the families of coordinate lines of curvature.
In this paper, the horizontal projection of the axis of the tubular surface is a logarithmic spiral. Parametric equations of the tubular surface in analytical form have been derived. A surface with the elevation angle of the axis β = 10° and the radius of the generating circle ρ = 15 linear units was constructed. The orthogonality of the resulting coordinate grid has been proven through the analysis of the coefficients of the first quadratic form (F = 0), which confirms the assignment of the surface to the lines of curvature. This makes it possible to improve the accuracy of calculating the stress-strain state of shells in mechanical engineering and aerospace engineering at simultaneous minimization of computational costs
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Copyright (c) 2026 Andrii Nesvidomin, Serhii Pylypaka, Tetiana Volina, Victor Nesvidomin, Oleksandr Solarov, Taras Voloshko, Taras Pylypaka, Lidiia Savchenko, Oleksandr Savchenko, Irina Zakharova
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