Construction of a mathematical model for approximating the sphere by strips of unfolding surfaces

Abstract

Approximating non-expandable surfaces by compartments of expandable ones makes it possible to simplify the process of obtaining the required shape without loss of operational properties. There is a known approximation of a sphere using the example of a ball when its surface can consist of polygons. However, this list does not exhaust the possible options for approximating the sphere. Its approximation by truncated cones tangent to parallels or by congruent cylindrical petals tangent to meridians is known. Any line on the surface of a sphere is a line of curvature. This means that the common line of contact of the expanded surface with the sphere will be a line of curvature for the expanded surface as well (rectilinear generatrices of the expanded surface will cross this line at a right angle). When building a sweep of such a surface, the line of contact will be transformed but the rectilinear generatrices will remain perpendicular to it, which simplifies the construction of the sweep. The approximation of the sphere by congruent strips, the number of which can be different, starting from one, is considered. A necessary condition for such an approximation is a common line of contact of adjacent strips. To this end, the line of contact on the sphere or the guide curve must have an appropriate shape. Such a curve is taken as a slope line (a curve whose tangents form a constant angle of inclination to the horizontal plane). The study results are the parametric equations of the strip touching the sphere and its corresponding equations on the sweep. The construction of the strip on the sweep is explained by the invariance of the geodesic curvature of the guide curve when the strip is bent until it aligns with the plane. This explains the difference between the proposed approach and conventional methods of sphere approximation. Approximating the sphere by strips of unfolding surfaces has a practical application in architecture with spherical elements, as well as in religious buildings with domes in the form of a part of the sphere