Computing spherical parameterizations for genus-zero closed surfaces is a fundamental task for geometric processing and computer graphics. Existing methods usually suffer from a lack of practical robustness or poor quality. In this paper, we present a practically robust method to compute high-quality spherical parameterizations with bijection and low isometric distortion. Our method is based on the hierarchical scheme containing mesh decimation and parameterization refinement. The practical robustness of our method relies on two novel techniques. The first one is a flat-to-extrusive decimation strategy, which contains two decimation error metrics to alleviate the difficulty of further mesh refinement. The second is a flexible group refinement technique that consists of flexible vertex insertion and efficient volumetric distortion minimization to control the maximum distortion. We convert the task of volumetric distortion minimization to one of tetrahedral mesh improvement to make the vertices distribute uniformly for efficient refinement. Compared with state-of-the-art methods, our method is more practically robust and possesses better mapping qualities. We demonstrate the efficacy of our method in spherical parameterization computations on a data set containing over five thousand complex models.

Frame shapes, which are made of struts, have been widely used in many fields, such as art, sculpture, architecture, and geometric modeling, etc. An interest in robotic fabrication of frame shapes via spatial thermoplastic extrusion has been increasingly growing in recent years. In this paper, we present a novel algorithm to generate a feasible fabrication sequence for general frame shapes. To solve this non-trivial combinatorial problem, we develop a divide-and-conquer strategy that first decomposes the input frame shape into stable layers via a constrained sparse optimization model. Then we search a feasible sequence for each layer via a local optimization method together with a backtracking strategy. The generated sequence guarantees that the already-printed part is in a stable equilibrium state at all stages of fabrication, and that the 3D printing extrusion head does not collide with the printed part during the fabrication. Our algorithm has been validated by a built prototype robotic fabrication system made by a 6-axis KUKA robotic arm with a customized extrusion head. Experimental results demonstrate the feasibility and applicability of our algorithm.