RELATION BETWEEN A HYPERBOLA AND AN ELLIPSE ON THE SURFACE OF A SPHERE

Abstract

Plane and spherical curves have common properties that are used in practice. Plane curves can slide as desired in the plane, making both translational and rotational movements in it. Similar movements can be made by spherical curves, moving from one position to another. For example, an analogue of a circle in a plane is also a circle on the surface of a sphere. Both curves are flat. An analogue of an ellipse in a plane is a spherical ellipse, which is a spatial curve, but the graphical methods of construction are common to both the plane and the surface of the sphere. These common geometric properties are used in the creation of spherical mechanisms, which are analogues of flat ones. For example, a pair of circles revolving around fixed centers and simultaneously sliding without slipping on each other is the basis for designing centroids for spur gears between parallel axes. The same circles on the surface of the sphere are the basis for designing centroids for bevel gears between axes that intersect at the center of the sphere. The formation of spherical curves is based on graphic constructions similar to curves on a plane. For example, an ellipse on a plane is formed as a locus of points, the sum of the distances from which to two given points is constant. A spherical ellipse is formed similarly, taking into account the fact that distances are measured on the surface of the sphere by arcs of great circles. It is convenient to take the radius of the sphere equal to one, then the distance on its surface is measured in angles. The difference between an ellipse and a hyperbola is that in the first case the sum of the distances is constant, and in the second – the difference. The given points are called the foci of the curves. If the foci of an ellipse and a hyperbola are common, then such curves are called confocal. By setting a constant distance between the foci and changing the sum or difference of their distances from the current point of the curve, families of confocal ellipses and hyperbolas can be obtained. They form an orthogonal grid both on the plane and on the sphere. The peculiarity is that on the sphere the analog of a plane hyperbola is a spherical ellipse, and two branches of the hyperbola on the plane correspond to two ellipses on the surface of the sphere. The article shows the relations between a hyperbola and an ellipse on the surface of a sphere. The peculiarity of this relationship is that the analog of a hyperbola on a sphere is a spherical ellipse. Parametric equations of confocal spherical hyperbolas and ellipses are derived. Orthogonal grids formed by confocal spherical ellipses and hyperbolas are built on the surface of the sphere.