Principles of Specifying a Program Trajectory by N-Focal Curves

The article proposes an approach to improving the use of isostages through their generalized representation when specifying a vessel’s program trajectory under conditions of restricted navigation. It is assumed that the principle of automatic tracking of multiple navigational marks by onboard navigational aids has been implemented. Based on previous research, the feasibility of approximating the boundaries of navigational hazards and the navigational safety domain by N-focal ellipses is substantiated. It has been shown that second-order N-focal curves allow for a more accurate approximation of the geometric shapes of navigational hazards without significant overlap of possible vessel maneuvering areas, while being directly associated with navigational parameters measured by onboard technical navigational aids and therefore independent of global satellite positioning systems. In this study, further development is proposed by representing isolines of navigational parameters using a generalized second-order N-focal curve. For this purpose, the concept of weight is introduced, which, under certain combinations, makes it possible to obtain an entire family of second-order curves (circles, ellipses, hyperbolas, N-focal ellipses, and N-focal hyperbolas) while keeping the initial measurements unchanged. Taking into account the specific arrangement of reference navigational marks, the affine transformation method is supplemented by variations in the base length between the reference marks. This approach makes it possible, for example, at the preliminary route planning stage, to obtain a complete grid of leading, clearing, and control isolines for a given program trajectory and its corridor in the form of second-order curves. The key results of the study are visualized, and the implementation of the proposed methods in modern navigation systems and autonomous vessel control systems will enable navigation information processing in coastal waters at a qualitatively new level.