СФЕРИЧНИЙ АНАЛОГ КОНХОЇДИ НІКОМЕДА

There exists a fundamental difference between planar figures and their spherical analogues. This can be demonstrated through the properties of plane and spherical triangles. For plane triangles, the sum of the internal angles equals 180°, while for spherical triangles this rule does not hold. However, following the same geometric construction principles used for planar curves—those based on the measurement of line segments—it is possible to create spherical analogues in which these segments correspond to the lengths of arcs of great circles. If such arcs are situated on a unit sphere, their lengths are numerically equal to the central angles subtended by radii connecting the sphere’s center to the arc endpoints. For example, a spherical analogue of the ellipse can be defined on the sphere by two fixed points such that the sum of the spherical distances from these points to any point on the curve is constant. Despite the differences between planar and spherical figures, they share several common properties. Just as plane figures can slide along the plane in all their points, spherical figures can similarly glide over the surface of the sphere. Some relationships between sides and angles that hold for plane figures are also valid for spherical ones. For instance, there exists a spherical analogue of the Pythagorean theorem, based on analogous relations between the legs and hypotenuse of a right triangle—both in the plane and on the sphere. The intersection of a sphere by a plane is a circle. Selecting three points on it, one can construct a spherical triangle with vertices at those points. Accordingly, for every spherical triangle there exists a circle passing through its vertices, just as for a planar triangle. If the triangle’s vertices lie on a great circle, this corresponds in the plane to vertices lying on a straight line, confirming that the spherical analogue of a straight line is a great circle. This paper considers the spherical analogue of Nicomedes’ conchoid. Its construction is based on laying off arcs of equal angular length in both directions from a great circle—the spherical counterpart of a straight line. These arcs are measured along great circles passing through a point located at a specified angular distance from the great circle. Analytical relationships describing this curve have been derived, and visualization of the obtained results has been performed.