Modeling of minimal parametrical networks in euclidean spaces by means of linkages

Abstract

Linkages can be represented as devices consisting of solid bodies, for example, rods, some pairs of which are connected to each other by hinges, in other words they have a common point
around which they can freely rotate. Linkages became widespread along with the development of instrumentation. One of the important first problems was to design a mechanism in which one of the hinges would move along a straight line segment. This issue has received several solutions, some of which were proposed by Peaucellier, Lipkin, Watt, Garth. After it became
clear how to draw a segment, the next big problem was to describe all possible curves that could be the trajectories of one of the hinges of a linkage. The solution to this problem was King’s theorem, which says that a set can be drawn if and only if it is either an ambient space or a semi-algebraic compact [16], [17].
The issues investigated by the author of this paper continue the exploration of previous tasks related to linkages, since they consider the possibilities of solving optimization problems
using linkages, for example, finding the shortest network connecting a set of points in Euclidean space. The main result of this work describes the construction of a mechanism that builds a
minimal parametric network in a Euclidean space of dimension 𝑑 > 2. In the author’s previous work, a proof of the existence of a linkages that builds a minimal Steiner network is given, and
a variant of constructing such a mechanism is also proposed. Since the main task was to prove the existence of such a mechanism, without minimizing it. The described assembly method can obviously be optimized and the results obtained in this work allows us to do that.