GEOMETRIC CONDITIONS FOR THE APPEARANCE OF DISCRETE ANALOGS OF SINGULAR POINTS ON DISCRETELY REPRESENTED CURVES

Abstract

The problems of creating the external forms of design objects are inextricably linked to the geometric modeling of objects, which includes the search for optimal forms of curved lines that would meet aesthetic, economic, compositional, geometric, static, and other requirements. Today, one of the most important tasks of applied geometry is to determine the geometric shape of a curved line, as well as the laws of its change and variation in shape. It is the geometric shape that can become a criterion for the aesthetics of the shape of a future product.

In discrete geometric modeling by the static-geometric method (SGM), the shape of curved lines depends on the external shaping load on the nodes of the discretely represented model and the presence of additional initial conditions, which, in turn, may contain hidden factors of operational control of the shaping process. The development of algorithms for constructing discrete models of curved lines, taking into account the peculiarities of the dependencies of the external load parameters, will allow solving the problems of the appearance of special points of curves and determining the criteria of aesthetics and expressiveness.

The research described in this paper is a continuation of previously presented studies on the search for discrete analogs of special points of curved lines depending on the parameters of the vertical external load on the nodes of the geometric model. The presented algorithms will allow us to control the shape of geometric models of curved lines or lines of surface contours. Due to this, the list of tasks for modeling discrete models of curved lines, finding various singular points on them, in particular, points of bending, straightening, and breaking, can be expanded. The conditions for the appearance of these points on discrete curves, as well as the identification and localization of discrete model nodes, will be a powerful tool for modifying the shape of curved objects. The presented studies demonstrate that when using (SGM), the main control parameters for varying the shape of curve models are the proportionality coefficient and the magnitude of the force vectors applied to the nodes of a discretely represented curved line.

Key words: singular points on curves; point of straightening; points of zero curvature; discrete analog; Inflection point on the curve; external forces; static geometric method.