Litnitskiy S. Approximation of discretely presented surfaces on the basis of their determinants approximate definition

Thesis is devoted to аpproximation of discretely presented surfaces on the basis of their determinants approximate definition by using differential geometric characteristics discretely analogs and optimization methods. Comparison of ways to define discretely analogs of differential geometric characteristics of curves and surfaces is made. Computational experiments are made and the most accurate and universal ways are chosen. These discretely analogs of differential geometric characteristics were used in solving problems process of research. The generatrix and discretely presented flat directing of discretely presented cylindrical surface approximate definition algorithm is made. The generatrix had minimal sum of scalar products of this generatrix and normals for discretely presented surface. Discretely presented flat directing was built on a plane perpendicular to the generatrix. The directing points ordering way was developed. The vertex and discretely presented flat directing of discretely presented conic surface approximate definition algorithm is made. The vertex had minimal sum of distances to generatrix vectors of conic surface. Generating vectors were founded by vector product calculation of two normals for discretely presented surface. Discretely presented flat directing was built on a plane perpendicular to the conic surface approximate generatrix. The directing points ordering way was developed. The rotation axis and discretely presented flat generatrix of discretely presented rotation surface approximate definition algorithm is made. The rotation axis had minimal sum of distances to normals for discretely presented surface. The plane on which lies the rotation axis was built. Discretely presented flat generatrix was created on that plane. The generatrix points ordering way was developed. The tangent developable described by the set of straight lines discretely presented space curve approximate definition algorithm is made. Perpendiculars between adjacent straight lines were built. Discretely presented space curve points were found near intersection points of these perpendiculars with straight lines. The algorithm of axis approximate definition of conoid described by the set of straight lines is made. The approximate conoid axis is a lane. Conoid is created by straight lines. The axis area possible location was founded. After that the approximate conoid axis with the minimal length to straight lines was defined. Accuracy and efficiency evaluation ways of the developed algorithms are proposed. The developed algorithms results were shown. Required discrete carcass was created on continual given surfaces with known equation for accuracy and efficiency evaluation of the developed algorithms. There was opportunity to compare approximate values with real one because the surfaces equations were known. Also there were discretely presented surfaces which points had some deviation. So these points did not lie on the necessary type surface. The discretely presented surfaces with these points were also analyzed. Ways to assess approximation accuracy of discretely presented surfaces are proposed and results of these ways were shown.