On the base of geometrical model of interior revolving one axoid by another for the pairs of contacted cylinders and cones analytical representations, and computer visualization of the new constructed kinematical surfaces is developed in this research. The described geometrical model corresponds to the case when one axoid is located in the interior of another axoid. As this takes place, there are two possible variants of mutual arrangement of moving and fixed axoids. In the first variant, the moving axoid is located in the interior of the fixed axoid. In the second variant, the fixed axoid is located in the interior of the moving axoid. In the first variant, the outside surface of the moving axoid revolves around the interior surface of the fixed axoid. In the second variant, the interior surface of the moving axoid revolves around the outside surface of the fixed axoid. For pairs of axoids “cylinder – cylinder” or “cone – cone” both variants of geometrical model for constructing kinematical ruled surfaces are considered on the base of rolling one axoid along another one. As a result of the moving one axoid along another the kinematical ruled surface is generated by one of the generating lines of moving axoid. Computer graphics of the new constructed kinematical ruled surfaces based on the proposed geometrical models has been accomplished by the previously developed software application “ArtMathGraph”.

Keywords: Mathematical Modeling, Analytical Geometry; Kinematical Ruled Surfaces, Computer Graphics

The geometrical model of interior revolving one axoid by another for the pairs of contacted one-sheet hyperboloids of revolution, as the methodical base for constructing of the new kinematical ruled surfaces is proposed in this article. Analytical representation of the constructed kinematical ruled surfaces is based on the previously developed model of complex moving one axoid along another. For the case of pair of contacted one-sheet hyperboloids of revolution the geometrical model of the complex moving can be represented as a superposition of some interrelated movements: rotational movement of the moving axoid around its axis; rotational movement of the moving axoid’s axis around the fixed axoid’s axis; translational movement of the moving axoid along the common generating line of both axoids. The proposed geometrical model of interior revolving one axoid by another corresponds to the case when one axoid is located in the interior of another axoid. As this takes place, there are two possible variants of mutual arrangement of moving and fixed axoids. In the first variant, the moving axoid is located in the interior of the fixed axoid. In the second variant, the fixed axoid is located in the interior of the moving axoid. In the first variant, the outside surface of the moving axoid revolves around the interior surface of the fixed axoid. In the second variant, the interior surface of the moving axoid revolves around the outside surface of the fixed axoid. As a result of the complex moving one axoid along another the kinematical surface is generated by one of the generating lines of moving axoid. Computer graphics of the new constructed kinematical ruled surfaces has been accomplished by the previously developed software application “ArtMathGraph”.

Keywords: Mathematical Modeling, Analytical Geometry; Kinematical Ruled Surfaces, Computer Graphics

  This paper presents the mathematical model for the construction of visual images of the intricate geometrical objects by means of computer graphics of transformed initial well-known surfaces. The transformed surfaces are a result of applying the mathematical transformations to the analytical representations of initial well-known classical surfaces such as plane, cone, cylinder, sphere, ellipsoid, etc. Graphical means of computer visualization of the geometrical objects is included  in the previously developed software application “ArtMathGraph” (AMG). The AMG application provides the possibility to interactive combine the individual images into computer compositions of derivable intricate geometrical objects as computer models of technical, natural or architectural visual images.    

Keywords: Mathematical Modeling, Analytical Surfaces, Computer Visualization

  New mathematical 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. On the base of this model analytical development and computer graphics of the new kinematic ruled surfaces are realized.   

Keywords: Mathematical Modeling, Computer Graphics, Kinematic Surfaces