A shift-invariant c2 1 the lattice-subdivision algorithm which rotates

In order to study differential properties of a subdivision surface at a markpoint, it is necessary to parametrise it over a so-called characteristic map defined as the infinite union of Ck-parametrised rings. Construction of this map is known when a single step of the subdivision scheme does not rotate a regular lattice. Otherwise, two steps are considered as they realign the lattice and its subdivided version. We present a new subdivision scheme which rotates the lattice and nevertheless allows a direct construction of the characteristic map. It is eigenanalysed with techniques introduced in a companion article and proved to define a C12-algorithm around a face-centre. This scheme generalises Loop's scheme, allowing the designer to choose between extraordinary vertices or faces in regard to the shape of the mesh, the location of the extraordinary elements, and the aimed limit shape.