THE FORMATION OF SPHERICAL EPICYCLOIDS WHEN THE MOVING CONE IS ROLLED ON A NON-MOVING CONE

Abstract

Many plane curves have spherical analog. They are united by the same methods of formation. One such example is the formation of an involute toothed surface. For a cylindrical transmission, the formation of the tooth surface occurs following a rectilinear generating cylinder that rolls along the plane. The orthogonal section of such surface is the involute of a circle. For a bevel transmission, the formation of the tooth surface occurs following a rectilinear generating cone, which also rolls along the plane. With such rolling, the top of the cone is stationary, and its base – a circle – forms a set of plane sections of the sphere with the center at the top of the cone. When rolling a cone, its base lies on the sphere with all its points, so a fixed point of the circle describes a spherical curve – an analogue of the involute of a circle on a plane. This curve can be obtained as a result of the intersection of a sphere with a conical surface, which is formed by a straight line of a cone rolling on a plane. Thus, the formation of a flat and spherical involute is similar, and in one case, a cylinder rolls along the plane, and in the other, a cone. There is also a spherical ellipse, the formation of which is similar to the formation of an ellipse on a plane. In both cases, it is a set of points, the sum of the distances from which to two fixed points is a constant value. The difference is that in one case the distances are measured by segments of straight lines, and in the other by arcs of circles, the radius of which is equal to the radius of the sphere. By analogy, spherical analogues of cycloids, hypo- and epicycloids can be constructed. An epicycloid is formed by the trail of a fixed point of a moving circle during its outward rolling along a fixed circle. Accordingly, for the formation of a spherical analogue of an epicycloid, it is necessary to consider the external rolling of a moving cone on a stationary one. The article provides an analytical description of such rolling, which is based on the fact that the bases of cones, which are circles, are located on the surface of a sphere. By analogy with the rolling of circles one by one in a plane, the rolling of circles one by one on the surface of a sphere is realized. The parametric equations of the spherical epicycloid with such rolling were obtained. In the partial case, when a fixed cone has an angle at the top equal to 180°, i.e. it turns into a plane, the spherical curve is the analogue of the cycloid on the plane. The obtained results were visualized using computer graphics.