USING SPATIAL PROJECTIVE TRANSFORMATION IN THE STATICS-GEOMETRIC METHOD

This is the first in a series of articles providing recommendations for generalizing the statics-geometric method and expanding its capabilities by eliminating the shortcomings of parabolic interpolation. It proposes using the simplest projective transformation, namely spatial perspective transformation, for modeling discrete frameworks. Research results suggest that the properties of the transformation significantly impact the process of discrete modeling of curved surfaces. By employing such a transformation, it becomes possible to model discrete frameworks of curved lines with specified vertical tangents. Alternatively, it allows for modeling discrete frameworks of surfaces with specified vertical tangent planes or specified vertical straight lines tangent to the surface. This transformation is one-to-one. We hope that the approach described in this work will allow for investigating the properties of modeled geometric objects based on initial conditions and transformation properties. The presented constructive algorithm visually demonstrates the features of spatial perspective transformation and will help construct discrete frameworks of complex architectural shells. It should also be noted that the use of direct and inverse transformation formulas in the process of discrete modeling can become a powerful tool for modeling surfaces with desired properties. The only crucial step is to establish rules for the correspondence between the modeled objects and the chosen transformation, and to define the transformation law for given points and lines. The application of spatial projective transformation in the form of spatial homology can serve as an effective means for generating new curved lines and surfaces, and for studying their properties