New geometrical model for constructing kinematical ruled surfaces on the base of interrelated movements in triads of contacting axoids, comprising of one fixed (1) axoid and two moving (2, 3) axoids, is proposed in this research. The methodical principles of this model is that the movement of axoid 2 along fixed axoid 1 defines such reconciled with it backward motion of axoid 3 along axoid 2, that the positional relationship of triad’s axoids is kept constant throughout the movement. As a result, the movement of one of rectilinear generators of moving axoid 3 generates a new kinematical ruled surface in the coordinate system, bound to fixed axoid 1. It’s been shown that the transition from well-known models of such contacting pairs of ruled surfaces as “plane – cylinder”, “plane – cone”, “cylinder – cylinder”, or “cone – cone” to models of interrelated movements in triads of contacting axoids opens up new opportunities for constructing various kinematical ruled surfaces. The corresponding analytical representation and computer visualization of generated kinematical ruled surfaces, based on triads of contacting axoids “plane – circular cylinder – circular cylinder”, “plane – circular cone – circular cone”, “circular cylinder – circular cylinder – circular cylinder”, and “circular cone – circular cone – circular cone”, has been developed. The parametric dependence of generated ruled surfaces on original ruled surfaces of the triad of contacting axoids offers a wide variety of resulting ruled surfaces. The proposed model with regard to graphic capabilities of the previously developed software application “ArtMathGraph” can be used as an instrument for computerized modeling technologically in-demand ruled surfaces.
geometrical modeling, analytical geometry, kinematical ruled surface, computer graphics.
Современные достижения геометрического моделирования аналитических поверхностей систематизированы в «Энциклопедии аналитических поверхностей» [12], включившей в себя, в частности, класс технологически востребованных линейчатых поверхностей [15; 16; 19]. Разработка новых геометрических моделей построения оригинальных аналитических поверхностей в сочетании с использованием современных технологий компьютерной графики [3; 7; 13] моделируемых поверхностей относится к одному из актуальных направлений аналитической геометрии линейчатых поверхностей [14; 17; 30], включая прикладные аспекты в строительстве и архитектуре [9; 10; 18].
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