Method of Rotation of Geometrical Objects Around the Curvilinear Axis

Abstract

Rotation is the motion of geometric objects along a circle. This is one of geometric techniques used to form lines and surfaces. In this paper has been considered the rotation of objects in a three-dimensional space around a straight axis. It is known that a straight line can be considered as a particular case of a circle with a radius equal to infinity. Such circle’s center is at infinite distance from the considered straight line segment. Then in the general case, the rotation axis is a closed curve, for example, a circle with a radius of finite magnitude. Rotation of a point around a straight axis now splits into two trajectories. One of them is a circle with a radius, the second is a straight line crossing with the axis, and the center of this trajectory is at an infinite distance from the point. The method of point rotation about an axis of finite radius was considered. Note that a circle is a special case of an ellipse. When the actual focus of the circle is stratified into two, the line itself loses its curvature constancy, and is called an ellipse. The point, rotating around the elliptical axis, is stratified into four ones, forming four circles (trajectories). Axis foci appearing in turn in the role of the main one determine two trajectories by each with a trivial and nontrivial center of rotation. We have considered the variant for arrangement of the generating circle so that its center coincided with one of the elliptic axis’s foci. The obtained surfaces are a pair of co-axial Dupin cyclides, since they have identical properties. Changing the circle generatrix radius, other things being equal, we get different types of closed cyclides.