О применении преобразований при решении задач начертательной геометрии

Abstract

This publication is devoted to the application of transformations at descriptive geometry’s problems solution. Using parametric calculus lets rationally select the number of transformations in the drawing. In Cartesian coordinates, on condition that an identical coordinate plane exists, the difference between parameters of linear forms, given and converted ones, is equal to the number of transformations in the composition. In affine space under these conditions, this difference is equal to two. Based on parameters calculation the conclusion is confirmed that the method of rotation around the level line, as providing the transformation of the plane of general position to the level plane, is a composition of two transformations: replacement of projections planes and rotation around the projection line. In various geometries (affine, projective, algebraic ones, and topology) the types of corresponding transformations are studied. As a result of these transformations are obtained affine, projective, bi-rational and topologically equivalent figures respectively. Such transformations are widely used in solving of applied problems, for example, in the design of technical surfaces of dependent sections. At the same time, along with transformation invariants, the simplicity of the algorithm for constructing of corresponding figures should be taken into account, with the result that so-called stratified transformations are preferred. A sign of transformation’s stratification is a value of dimension for a set of corresponding points’ carriers. This fact explains the relative simplicity of the algorithm for constructing the corresponding points in such transformations. In this paper the use of stratified transformations when finding the points of intersection of a curve with a surface, as well as in the construction of surfaces with variable cross-section shape are considered. The given examples show stratification idea possibilities for solving the problems of descriptive geometry.