CONGRUENT AXOIDS OF NON-CIRCULAR CONICAL WHEELS FORMED BY ARCS OF THE LOXODROME

If two linear surfaces move one after another without sliding, then they can be considered as axoids of a solid body that performs a corresponding motion in space. If the axoids are cylindrical surfaces, then the study of their rolling can be replaced by the study of the rolling of centroids - curves of the orthogonal cross section of these cylindrical surfaces. Usually the rolling of a moving centroid on a stationary one is considered. However, there are cases when centroids roll one after another, while rotating around fixed centers. Examples of round centroids are circles, non-round - congruent ellipses, in which the centers of rotation are foci. In both cases, the center-to-center distance is constant. The point of contact of the circles is located at the center distance and is stationary during their rotation, and for ellipses it "floats" on this segment. In the article [1] the congruent centroids formed by symmetric arcs of a logarithmic spiral are considered. The centers of rotation of the centroid are the poles of the spirals. A characteristic feature of logarithmic spirals is that they intersect all radius vectors emanating from the pole at a constant angle. For a sphere, the prototype of a logarithmic spiral is a loxodrome, which crosses all the meridians at a constant angle and twists around the pole of the sphere. The paper hypothesizes that closed spherical curves formed from arcs of loxodrome, like centroids from arcs of a logarithmic spiral on a plane, can also roll around axes intersecting in the center of the sphere. If these closed curves are connected by rectilinear segments with the center of the sphere, then two cones are formed - axoids of non- circular conical wheels. This hypothesis is based on the fact that with an infinite increase in the radius of the sphere, its surface around the pole turns into a plane, and the meridians - in straight lines emanating from the pole. Accordingly, the loxodrome is transformed into a logarithmic spiral. The article shows that the hypothesis is confirmed. Axoids of non- circular wheels are built in it, the axes of which intersect at right angles. The expression of the arc length of the loxodrome is found and it is shown that when the conical axoids rotate at appropriate angles around their axes, the contact curves pass straight paths. This means that the rolling of the axis occurs without sliding.