07: Curved Crease Edge Rounding of Polyhedral Surfaces - Maleczek et al

Описание: Authors
R. Maleczek, K. Mundilova, T. Tachi

Abstract
We show a novel method to design a curved crease folding that  constructs the edgerounded, i.e., filleted, version of a given polyhedral surface. We replace each edge with a smoothly rounded cylinder and each vertex with a generalised cone, such that the surfaces joined through curved creases form a single developable surface with possible cuts at the singular cone apices. Because the curved crease can be explicitly computed from the isometry of corresponding line segments for given locations of the cone apex in 2D and 3D, our problem reduces to identifying the locations of the apices. We characterise the conditions for the apex positions and provide a numerical scheme to find the apices for the given mesh by solving a nonlinear optimisation problem. In general, the rounding of edges reduces the surface area, so the resulting curved folded surface is not isometric to the original polyhedron; in particular, the surface is not guaranteed to be foldable from a single piece of uncut paper if applied to a developable polyhedral surface. We solve this problem by  computing consistent material loss caused by rounding radii.