Construction of canal surfaces based on a specified flat curvature line

Abstract

The research relevance is predefined by the widespread use of elements and structures that have the shape of canal surfaces in engineering practice and the possibilities of reproducing the surface through kinematics. The research aims to develop new means of modeling canal surfaces referred to as a grid of curvature lines by introducing elements with special properties into the structural model that simplify the solution of differential equations and reduce the amount of computation. To achieve the research methods, the synthetic geometric method, methods of linear algebra, the theory of differential equations and differential geometry, as well as methods of computer geometric modeling and visualization of three-dimensional objects were used. Studies on modeling and studying the properties of channel surfaces are analyzed. The research on the problem of the surfaces and lines of curvature is considered in more detail and the conditions under which it is possible to simplify the solution of the differential equation are identified. It was proved that the condition of contact between the canal surface and the plane along a given plane curve is sufficient for this curve to be one of the curvature lines of the family of orthogonal to the generating circles. This allowed to reduce the solution of the differential equation to two quadratures. The expressions of the corresponding integrals and an algorithm for modeling the canal surface with the possibility of referring to a grid of curvature lines were obtained. The expressions that define the desired surface include the parametric equations of a given plane line; a function that determines the radii of the spheres of the family depending on the parameter of this line. A specific example of modeling a surface based on a defined formula was also considered, and images of this surface with visualization of the coordinate grid were presented. The research’s practical values are defined by the possibility of using the developed modeling tools in the design and computer-aided design of the geometry of real products containing surfaces of a smooth transition of variable radius