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Mathematical models for solving optimization problems of connection in non-simply connected domains with typical technological constraints on the geometric and topological parameters of the routes, first of all, on the curvature and the number of kinks, are considered and developed. These models are combined with existing and prospective topogeodetic models of the polygonal image of territories. The solution of connection problems is associated with the search for optimal trajectories of traces and networks in sections of free geometric shape, which requires the development of fairly general models as areas in which these connections are realized. These can be junction types such as polyline, Manhattan, smooth, solid, and other types of traces. As shown in the works of SmelyakovS.V. and Aliseyko A.A. (PlekhovaA.A.) global and local regularization of geometric constructions in solving connection problems [1], the general optimization problem of connections can be formulated as the problem of choosing , where – is a set of alternatives, and is the optimality principle. In this case, the set – can be represented as a set of of the phase space and the constraints imposed on the parameters of the phase space . In turn, the phase space is expedient to represent the Cartesian product of the initial data , disturbances , control parameters and results . As the analysis of the problem [1] shows, the efficiency of modeling the phase space is primarily related to the description of the initial data in the section and the space of admissible traces in . This issue is considered as the development of the construction of the structure of models and the methodology of their use, which allows the possibility of constructive and efficient (computationally) construction and enumeration of various models and algorithms that preserve the geometric invariance of the models required for a specific use under the conditions of admissibility of using various structures of the initial data. This work is devoted to solving the problem of creating such a model within the boundaries of geometric design for connection problems.