Formalization of technological processes based on non-Euclidean geometries: spherical and Riemann geometry

It is a continuation of the set of research works [1]–[4] on modeling and formalization in the direction of ideas and concepts of the geometries mentioned above. First of all, it concerns the technological interpretation of geometric images on the sphere, in particular, the elliptical plane. Intersecting or converging lines were studied here - this is a bundle of straight lines with its own vertex or an elliptical bundle. At the same time, at the macro level - in relation to TP, as a set of stages or states of ADS [2], and at the subordinate meso level - to a separate TP stage. In modeling, the surface of the sphere is considered as a spatial analogue of the central great circle (equator), the center of which coincides with the center of the sphere. The “field of action” of these geometries (as well as TP) is the surface of the sphere. And an important transformation of similarity in the theory is considered that the main “constructive” (forming) element is a now connected (glued) pair of diametrically opposite points (or points – antipodes [19] – a conditional “point”. With these antipodal points. The concepts of “line” and “plane” are defined precisely in Riemann’s geometry and in their subsequent technological interpretation. is a set of great circles (TP stages) and in spherical geometry it is a line, then in similarity transformations it is a “line” (a set of conditional “points”) placed on an elliptic plane. Since the elliptic space within the geometry of position contains at least four points, then they are given a technological interpretation with a formal presentation (quadratic form).