Abstract 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.

中文翻译：

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将球体划分为等面积和/或等边球面多边形

摘要 将球体均匀地划分为等面积或等边球面多边形对于各种实际应用非常有用。然而，实现如此均匀的球体细分是一项具有挑战性的任务。本研究通过数值近似研究了两类球体细分：（i）将球体划分为等面积的球面多边形；(ii) 将球体划分为所有边均具有单一长度的球面多边形。开发了一种计算工作流程，该工作流程在选定的案例研究中被证明是有效的。首先，根据最初由球形四边形组成的球体获得细分。允许在初始段内创建新顶点以生成子组件。这种方法提供了新的机会，可以通过初始段内分布点的自由移动来控制生成的细分球面多边形的面积和边长，而不限制边界点。在这项工作中提供了一系列示例，以证明所提出的方法可以有效地获得一系列等面积或等边球面四边形细分。发现在初始细分段之间创建间隙可以生成等边球面四边形。其次，本研究检查了等面积和/或等边长的球形五边形和戈德堡多面体细分。在球形五边形细分中，不需要球体上​​的间隙来实现相等的边长。除了，在球形 Goldberg 多面体细分中获得等面积或等边几何形状有很大的灵活性。第三，本研究发现了两个新颖的 Goldberg 球面细分，它们同时表现出相等的面积和相等的边长。