Investigation of the algorithm for constructing smooth spatial curves with the ability to specify the curvature and torsion at the nodal points

Abstract

In the existing methods for describing spline vector-parametric surfaces, a rich arsenal of algorithms has been accumulated. Often the development of hardware equipment outperforms the software. In addition, the state of the art often throws new «challenges»to the developer, for example, the need to obtain a channel surface with special properties, with predefined curvature and torsion, with a certain order of smoothness. This is important in the design of product pipelines, exhaust manifolds of internal combustion engines, etc. Therefore, it makes sense to investigate previously unexplored types of control over the future properties of spline curves, and surfaces based on them. Special polynomial splines of higher degrees (seventh degree) and mathematical apparatus (linear algebra, Cramer method, vector-parametric description of curves) are used in the work.

The formula for calculation of the vector-parametric segment of the seventh degree (on two end points, two first, second and third derivatives in them) is derived. The formula allows more flexible control over the shape of the desired spline, arbitrarily setting the initial data.

The algorithm given in the work allows to answer the «call», giving the designer complete freedom in controlling the geometric properties of the object even at the development stage. The algorithm allows to create better samples of modern technology thanks to the control of curvature and torsion.