Dividing a sphere into equal-area and/or equilateral spherical polygons

Dividing a sphere uniformly into equal-area or equilateral spherical polygons is useful for a wide variety of practical applications. However, achieving such a uniform subdivision of a sphere is a challenging task. This study investigates two classes of sphere subdivisions through numerical approximation: (i) dividing a sphere into spherical polygons of equal area; and (ii) dividing a sphere into spherical polygons with a single length for all edges. A computational workflow is developed that proved to be efficient on the selected case studies. First, the subdivisions are obtained based on spheres initially composed of spherical quadrangles. New vertices are allowed to be created within the initial segments to generate subcomponents. This approach offers new opportunities to control the area and edge length of generated subdivided spherical polygons through the free movement of distributed points within the initial segments without restricting the boundary points. A series of examples are presented in this work to demonstrate that the proposed approach can effectively obtain a range of equal-area or equilateral spherical quadrilateral subdivisions. It is found that creating gaps between initial subdivided segments enables the generation of equilateral spherical quadrangles. Secondly, this study examines spherical pentagonal and Goldberg polyhedral subdivisions for equal area and/or equal edge length. In the spherical pentagonal subdivision, gaps on the sphere are not required to achieve equal edge length. Besides, there is much flexibility in obtaining either the equal area or equilateral geometry in the spherical Goldberg polyhedral subdivisions. Thirdly, this study has discovered two novel Goldberg spherical subdivisions that simultaneously exhibit equal area and equal edge length.