Constructing geometrical models of spherical analogs of the involute of a circle and cycloid

Abstract

The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid