Abstract The focus is on circular nets with one or two families of spherical parameter lines, which are treated in Möbius geometry. These circular nets provide a discretisation of surfaces with one or two families of spherical curvature lines. The special cases of planar, circular and linear parameter lines are also investigated. A Lie-geometric discretisation in terms of principal contact element nets is also presented. Its properties are analogous to the classical properties of surfaces with one or two families of spherical curvature lines. Circular nets with two families of spherical parameter lines have geometric properties that are related to Darboux cyclides. Circular nets with one or two families of spherical parameter lines are examples of Q-nets with terminating Laplace sequences. More generally, this article considers Q-nets that are inscribed in quadrics and that have terminating Laplace sequences.

Abstract We introduce decorated piecewise hyperbolic and spherical surfaces and discuss their discrete conformal equivalence. A decoration is a choice of circle about each vertex of the surface. Our decorated surfaces are closely related to inversive distance circle packings, canonical tessellations of hyperbolic surfaces, and hyperbolic polyhedra. Discrete conformal equivalence introduces an equivalence relation on decorated piecewise hyperbolic and spherical surfaces. We prove the corresponding uniformization theorem—in each equivalence class there is a unique representative on a constant curvature surface. Furthermore, we introduce a fundamental invariant characterizing the discrete conformal classes. We show that one can deform continuously between decorated piecewise hyperbolic, Euclidean, and spherical surfaces sharing the same invariant. Our approach is based on a variational principle, which also provides a way to compute the discrete uniformization and geometric transitions.

Abstract In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.

A simplicial surface provides its carrier with a natural triangulation whose vertex set includes the cone points and the corners of the boundary. However, this triangulation is not intrinsically distinguished from other triangulations with the same vertex set, it is not preserved under isometric deformations of the surface. Delaunay tessellations of polyhedral surfaces are defined intrinsically in terms of empty discs on surfaces. The edges of Delaunay tessellations are geodesics on the original polyhedral surface (and not necessarily straight edges in the 3-space). For any polyhedral surface there exists a unique Delaunay tessellation. It is not necessarily strongly regular, i.e. the intersection of two closed cells may not be a single closed cell.

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node geometry, and multilayer constructions. Beams of constant height are achieved with the new type of edge offset meshes. Mesh parallelism is also the main ingredient in a novel discrete theory of curvatures. These methods are applied to the construction of quadrilateral, pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces, and their geometric transforms. We show how to design geometrically optimal shapes, and how to find a meaningful meshing and beam layout for existing shapes.