Geometric modeling and computer graphics of kinematic ruled surfaces on the base of complex moving one axoid along another (one-sheet hyperboloid of revolution as fixed and moving axoids)

Abstract

In concordance with logical structure “Geometrical Model of Interaction of two contacted surfaces during the movement of one ruled surface along another – Corresponding Mathematical Transformations of Surfaces – New Kinematic Ruled Surfaces” [1-3] a new geometrical model of complex moving one axoid along another for the case of one-sheet hyperboloid of revolution as fixed and moving axoids has been proposed. The main condition of constructing kinematic ruled surfaces is that moving axoid contact with fixed axoid along one their common generating line in each of their positions during complex moving one axoid along another. A case when the axes of fixed and moving axoids are crossed (Fig. 1,2,3), has been considered in this research. Analytical development and computer graphics of the new kinematic surfaces are realized for three types of complex moving. (1)The outside surface of the fixed axoid is revolved slipping-free by the outside surface of the corresponding moving axoid (Fig. 1). (2)The interior surface of the fixed axoid is revolved slipping-free by the outside surface of the corresponding moving axoid (Fig. 2). (3)The outside surface of the fixed axoid is revolved slipping-free by the interior surface of the corresponding moving axoid (Fig. 3). Computer graphics of the constructed surfaces (Fig. 1a,2a,3a) have been performed by the previously developed software application.