CHoCC: Convex Hull of Cospherical Circles and Applications to Lattices

Abstract We discuss the properties and computation of the boundary B of a CHoCC (Convex Hulls of Cospherical Circles), which we define as the curved convex hull H ( C ) of a set C of n oriented and cospherical circles { C i } that bound disjoint spherical caps of possibly different radii. The faces of B comprise: n disks, each bounded by an input circle, t = 2 n − 4 triangles, each having vertices on different circles, and 3 t ∕ 2 developable surfaces, which we call corridors. The connectivity of B and the vertices of its triangles may be obtained by computing the Apollonius diagram of a flattening of the caps via a stereographic projection. As a more direct alternative, we propose a construction that works directly in 3D. The corridors are each a subset of an elliptic cone and their four vertices are coplanar. We define a beam as the convex hull of two balls (on which it is incident) and a lattice as the union of beams that are incident each on a pair of balls of a given set. We say that a lattice is clean when its beams are disjoint, unless they are incident upon the same ball. To simplify the structure of a clean lattice, one may union it with copies of the balls that are each enlarged so that it includes all intersections of its incident beams. But doing so may increase the total volume of the lattice significantly. To reduce this side-effect, we propose to replace each enlarged ball by a CHoCC and to approximate the lattice by an ACHoCC, which is an assembly of non-interfering CHoCCs for which the contact-faces are disks. We also discuss polyhedral approximations of CHoCCs and of ACHoCCs and advocate their use for processing and printing lattices.