The Spherical Quad

Planar quadrilateral meshes are the workhorse of architectural geometry — every face lies in a plane, which means every panel can be manufactured flat. But the requirement of planarity constrains the shapes you can approximate. Laguerre geometry offers a different primitive: instead of planes meeting at edges, you get spheres meeting at conical strips, with planar quads only at vertices.

The resulting L-meshes are watertight surfaces where each face is a sphere of specified radius, each edge is a cone, and each vertex is a flat quad. In the continuous limit, this yields Laguerre conjugate nets — the analog of classical conjugate nets but in a geometry where spheres replace points and oriented contact replaces intersection.

The practical value is in the specification. Classical planar-quad meshes let you control face planarity but not curvature. L-meshes let you prescribe the radius of every spherical face — you’re not fitting planes to a curved surface but building the surface from spheres of chosen sizes. The curvature information is built into the mesh elements rather than emerging as an approximation artifact.

The theoretical value is structural. Laguerre geometry treats oriented planes and spheres as equivalent objects, connected by a transformation that maps incidence to tangency. Translating classical discrete differential geometry into this framework doesn’t just add spheres to the toolkit — it reveals which properties of planar-quad meshes were geometric necessities and which were artifacts of the Euclidean setting. Some constraints survive the translation. Others dissolve, replaced by genuinely different conditions that happen to look similar.