ON A METHOD OF DETERMINING INITIAL CONDITIONS FOR CONSTRUCTING A CHEBYSHEV NETWORK MODEL ON NON-DEVELOPABLE LINEAR SURFACES

Abstract

The article considers a method for constructing a geodesic line on a section of a non-expandable ruled surface. The first point of the curve is proposed to be chosen on the boundary of the surface section. The initial direction of construction is determined by a vector located in the plane tangent to the surface at the first point. The constructed geodesic is considered as the initial line of the first family of Chebyshev grid curves. Since when dressing a surface with a fabric with a mesh structure, its threads when applied to the surface are laid along the lines of the Chebyshev grid, it is proposed to choose the step of constructing the curve points taking into account the properties of both the surface itself and the fabric being dressed. Choosing the maximum permissible step size allows minimizing the number of constructions, while ensuring the absence of folds.

To find the points of the first geodesic, a method of sequential reprojection is proposed, based on the property of parallel transfer of vectors on the surface, according to which, with such transfer on the surface along the curve of two vectors that originate from one point Mi and belong to the surface to a nearby point Mi+1, the lengths of the vectors and the angle between them do not change. The construction of the geodesic ends when reaching the boundary of a given compartment of the lined surface. To select the first direction vector (a_0 ) ⃗ of the second family of grid lines, it is proposed to use the line of intersection of the plane tangent to the surface at the first point of the curve located on the boundary of the compartment and the normal plane, which is given by the normal vector to the surface and the initial direction of the curve. The construction of the following directions of vectors (a_i ) ⃗ is proposed to be implemented using an algorithm in which the constructed geodesic curve is considered as a striction line of a lined surface, which is given by the family of direction vectors (a_i ) ⃗ . This approach allows, with the help of a relatively small number of calculations, to obtain the initial conditions for constructing a Chebyshev grid model on a section of an unexpanded lined surface.

Keywords: geodesic line; Chebyshev grid; geodesically parallel vector transfer on a surface; surface dressing with a fabric on a mesh base; cell length of the Chebyshev grid model; striction line.