Moving and fixed axoids of frenet thrihedral of directing curve on example of cylindrical line

Abstract

If the solid body makes a spatial motion, then at any point in time this motion can be decomposed into rotational at angular velocity and translational at linear velocity. The direction of the axis of rotation and the magnitude of the angular velocity, that is the vector of rotational motion at a given time does not change regardless of the point of the solid body (pole), relative to which the decomposition of velocities. For linear velocity translational motion is the opposite: the magnitude and direction of the vector depend on the choice of the pole. In a solid body, you can find a point, that is, a pole with respect to which both vectors of rotational and translational motions have the same direction. The common line given by these two vectors is called the instantaneous axis of rotation and sliding, or the kinematic screw. It is characterized by the direction and parameter - the ratio of linear and angular velocity. If the linear velocity is zero and the angular velocity is not, then at this point in time the body performs only rotational motion. If it is the other way around, then the body moves in translational manner without rotating motion. The accompanying trihedral moves along the directing curve, it makes a spatial motion, that is, at any given time it is possible to find the position of the axis of the kinematic screw. Its location in the trihedral, as in a solid body, is well defined and depends entirely on the differential characteristics of the curve at the point of location of the trihedral – its curvature and torsion. Since, in the general case, the curvature and torsion change as the trihedral moves along the curve, then the position of the axis of the kinematic screw will also change. Multitude of these positions form a linear surface - an axoid. At the same time distinguish the fixed axoid relative to the fixed coordinate system, and the moving - which is formed in the system of the trihedral and moves with it. The shape of the moving and fixed axoids depends on the curve. The curve itself can be reproduced by rolling a moving axoid over a fixed one, while sliding along a common touch line at a linear velocity, which is also determined by the curvature and torsion of the curve at a particular point. For flat curves, there is no sliding, that is, the movable axoid is rolling over a stationary one without sliding. There is a set of curves for which the angular velocity of the rotation of the trihedral is constant. These include the helical line too. The article deals with axoids of cylindrical lines and some of them are constructed.