COMPUTER DESIGN OF ROUTES OF LINEAR STRUCTURES USING ALGORITHMS SPLINE-APPROXIMATIONS OF MULTIPLE-VALUED FUNCTIONS. PART 1

Abstract

The problem of converting a flat broken line into a spline, whose elements are arcs of circles conjugated by line segments, is considered. This problem arises when designing the plan and profile of the routes of linear structures. For such structures as trenches for laying pipelines for various purposes, it has independent significance, as well as in the design of the longitudinal profile of railways and highways. When designing a road alignment plan, the solution of this problem is considered as the first step in solving a more complex problem in which clothoids are used to pair arcs of circles and straight lines. With regard to the design of the longitudinal profile of roads, the problem of approximation by a spline with circles was solved earlier. The principal feature of the problem under consideration in relation to the design of the route plan is that the spline is a multivalued function and the previously developed algorithm is unsuitable. The problem is considered in the presence of restrictions on the parameters of the spline and an unknown number of its elements. It is proposed at the first stage to determine the number of spline elements, and at the second stage to optimize its parameters. As an objective function, a simple or weighted sum of squared deviations of the initial points (polyline nodes) from the spline is used. The article considers the first stage, taking into account the features of the approximation of multivalued functions. Implemented dynamic programming algorithm; the result is an initial guess for the non-linear programming algorithm. Fundamentally new is the calculation of the derivatives of the objective function with respect to the parameters that determine the position of the spline in the absence of its analytical expression through these parameters. The next article will be devoted to the solution of the problem at the second stage.