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.

Discrete surfaces with spherical faces are interesting from a simplified manufacturing viewpoint when compared to other double curved face shapes. Furthermore, by the nature of their definition they are also appealing from the theoretical side leading to a Möbius invariant discrete surface theory. We therefore systematically describe so called sphere meshes with spherical faces and circular arcs as edges where the Möbius transformation group acts on all of its elements. Driven by aspects important for manufacturing, we provide the means to cluster spherical panels by their radii. We investigate the generation of sphere meshes which allow for a geometric support structure and characterize all such meshes with triangular combinatorics in terms of non-Euclidean geometries. We generate sphere meshes with hexagonal combinatorics by intersecting tangential spheres of a reference surface and let them evolve - guided by the surface curvature - to visually convex hexagons, even in negatively curved areas. Furthermore, we extend meshes with circular faces of all combinatorics to sphere meshes by filling its circles with suitable spherical caps and provide a remeshing scheme to obtain quadrilateral sphere meshes with support structure from given sphere congruences. By broadening polyhedral meshes to sphere meshes we exploit the additional degrees of freedom to minimize intersection angles of neighboring spheres enabling the use of spherical panels that provide a softer perception of the overall surface.