Meshes with spherical faces and circular edges are an attractive alternative to polyhedral meshes for applications in architecture and design. Approximation of a given surface by such a mesh needs to consider the visual appearance, approximation quality, the position and orientation of circular intersections of neighboring faces and the existence of a torsion free support structure that is formed by the planes of circular edges. The latter requirement implies that the mesh simultaneously defines a second mesh whose faces lie on the same spheres as the faces of the first mesh. It is a discretization of the two envelopes of a sphere congruence, i.e., a two-parameter family of spheres. We relate such sphere congruences to torsal parameterizations of associated line congruences. Turning practical requirements into properties of such a line congruence, we optimize line and sphere congruence as a basis for computing a mesh with spherical triangular or quadrilateral faces that approximates a given reference surface.

中文翻译：

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通过具有球面的网格进行近似


对于建筑和设计中的应用，具有球面和圆形边的网格是多面体网格的有吸引力的替代方案。通过这种网格对给定表面进行近似时，需要考虑视觉外观、近似质量、相邻面的圆形交点的位置和方向，以及是否存在由圆形边平面形成的无扭支撑结构。后一个要求意味着网格同时定义第二个网格，其面与第一个网格的面位于相同的球体上。它是球体全等的两个包络线的离散化，即双参数球体族。我们将这种球体同余与相关线同余的扭转参数化联系起来。我们将实际要求转化为这种线同余的属性，优化了线和球面的同余，作为计算具有近似给定参考表面的球形三角形或四边形面的网格的基础。