Topology-preserving polyhedral design using ribbon-based multi-sided patches

Polyhedral design is a well-established paradigm for creating complex free-form shapes by smoothing control polyhedra. Several approaches exist, including recursive subdivision and direct algorithms. We introduce a direct method where multi-sided and multi-connected, ribbon-based patches are stitched together. While traditional methods are based on control polyhedra with only convex faces, our approach can handle concave angles and multiple hole loops, as well; moreover, the free-form patchwork has exactly the same topological structure as the input control polyhedron. The algorithm consists of three basic steps: (i) computing an auxiliary polyhedron, (ii) defining a general topology curve network and determining cross-derivative (ribbon) data along the patch boundaries, (iii) computing Generalized B-spline surfaces. Furthermore, using a shape parameter, a family of models with the same topological structure can be generated. Several test examples are provided to demonstrate the capabilities of this new method.