Reshaping Convex Polyhedra

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings. This investigation uncovered previously unexplored topics, including a notion of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto P. In the second part of this monograph, we study vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to produce non-overlapping polyhedral and planar unfoldings, which led us to develop an apparently new theory of convex sets, and of minimal length enclosing polygons, on convex polyhedra. All our theorem proofs are constructive, implying polynomial-time algorithms.Comment: Research monograph. 234 pages, 105 figures, 55 references. arXiv admin note: text overlap with arXiv:2008.0175

Gathering of Mobile Robots with Defected Views

An autonomous mobile robot system consisting of many mobile computational entities (called robots) attracts much attention of researchers, and it is an emerging issue for a recent couple of decades to clarify the relation between the capabilities of robots and solvability of the problems. Generally, each robot can observe all other robots as long as there are no restrictions on visibility range or obstructions, regardless of the number of robots. In this paper, we provide a new perspective on the observation by robots; a robot cannot necessarily observe all other robots regardless of distances to them. We call this new computational model the defected view model. Under this model, in this paper, we consider the gathering problem that requires all the robots to gather at the same non-predetermined point and propose two algorithms to solve the gathering problem in the adversarial (N,N-2)-defected model for N ? 5 (where each robot observes at most N-2 robots chosen adversarially) and the distance-based (4,2)-defected model (where each robot observes at most two robots closest to itself), respectively, where N is the number of robots. Moreover, we present an impossibility result showing that there is no (deterministic) gathering algorithm in the adversarial or distance-based (3,1)-defected model, and we also show an impossibility result for the gathering in a relaxed (N, N-2)-defected model

Conforming restricted Delaunay mesh generation for piecewise smooth complexes

A Frontal-Delaunay refinement algorithm for mesh generation in piecewise smooth domains is described. Built using a restricted Delaunay framework, this new algorithm combines a number of novel features, including: (i) an unweighted, conforming restricted Delaunay representation for domains specified as a (non-manifold) collection of piecewise smooth surface patches and curve segments, (ii) a protection strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of off-centre refinement rules designed to achieve high-quality point-placement along embedded curve features. Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl