The dissertation on competition of a scientific degree of a Cand.Tech.Sci. on a speciality 05.01.01-applied geometry, the engineering drawing. The Kiev national university of building and architecture, Kiev, Ukraine, 2011 The dissertation is devoted to the decision of an actual problem of geometrical modelling form-building covers of not developed surfaces made of composite materials on a fabric basis. For the purpose of creation independent of ways of the task of surfaces of algorithms and surface programs were set by the ordered skeleton of points with the subsequent analytical description surfaces of Coons of the second and fifth orders. So at clothing of a surface by a fabric of its thread settle down on lines of families of grid of Chebyshev, for modelling of position of a fabric on a surface researches of conditions of existence of grids of Chebyshev on any surfaces and those their properties which would allow with sufficient accuracy to build them on a surface have been executed. Such there was a following property: tangents vectors along lines of one family in points of their crossing with lines of the second family are transferred in parallel on surfaces and this property is mutual .In this work two ways of construction of in parallel transferred vector along a curve on a surface have been fulfilled. The first way is based on that fact, that parallel transferring of vectors on a surface concerns internal geometry of a surface Therefore it is enough to transfer a vector in parallel in usual sense on development of the torso, concerning surfaces along a transferring line, and the subsequent preservation of corners, created by a vector with tangents to a curve on development, with tangents to a curve on a surface. For construction of a torso, a direction which forming is interfaced to directions of tangents to a curve, curve display has been used. The second way was based on the analytical theory of parallel transferring of the vector in parallel transferred from a point of M in infinite point М1, is equal to zero, and full differential Dа is directed on a normal to a surface. Considering this fact, has been developed and according to a programm of the algorithm of definition of a direction of in parallel transferred vector аi from point Мi in point Мi+1, as result of crossing of the normal plane defined by a vector ai and a normal to a surface in point Mi and a tangent of a plane to a surface in point Mi+1 is realised. For practical construction of approached grid of Chebyshev on not developed surfaces the formulas establishing dependence between length of the side of a cell, have been deduced by curvature of a surface and properties of a fabric. Performance of these dependences allows is guaranteed to build the fourth top of a cell at three set. As parallel transferring of a vector on a surface is carried out not in infinitely close point, but in a point removed for the set length of the side of a cell is occurred the accumulation of errors and a deviation of lengths of the sides of cells from the set size. Therefore algorithms of updating of the side of a cell of a grid in the course of its construction have been developed. The researches, concerning constructions uniform of Chebyshev's grids of the next sites of the surface described by surfaces of Coons are executed also. As an example, the model Chebyshev's grids has been constructed on a surface of cowl of the keel of a plane and on its basis is constructed the development coverings of fiber glass fabric. The results of work in the form of programs and instructions to them are transferred to the state enterprise "Kiev aviation factory "Aviant" for introduction in manufacture. Keywords: parallel transferring of a vector, grid of Chebyshev, a cell, glass fabric, filling.
