A design tool for globally developable discrete architectural surfaces using Ricci flow

This paper presents an approach for the design of discrete architectural surfaces that are globally developable; that is, having zero Gaussian curvature at every interior node. This kind of architectural surface is particularly suitable for fast fabrication at a low cost, since their curved geometry can be developed into a plane. This highly non-linear design problem is broken down into two sub-problems: (1) find the member lengths of a triangular mesh that lead to zero Gaussian curvature, by employing the discrete surface Ricci flow developed in the field of discrete differential geometry; (2) realize the final geometry by solving an optimization problem, subject to the constraints on member lengths as well as the given boundary. It is demonstrated by the numerical examples that both of these two sub-problems can be solved with small computational costs and sufficient accuracy. In addition, the Ricci flow algorithm has an attractive feature-the final design is conformal to the initial one. Conformality could result in higher structural performance, because the shape of each panel is kept as close as possible to its initial design, suppressing possible distortion of the panels. This paper further presents an improved circle packing scheme implemented in the discrete surface Ricci flow to achieve better conformality, while keeping its simplicity in algorithm implementation as in the existing Thurston's scheme

Scalable and Fully Distributed Localization in Large-Scale Sensor Networks

This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and a non-uniform nodal distribution. In contrast to current state-of-the-art connectivity-based localization methods, the proposed algorithm is highly scalable with linear computation and communication costs with respect to the size of the network; and fully distributed where each node only needs the information of its neighbors without cumbersome partitioning and merging process. The algorithm is theoretically guaranteed and numerically stable. Moreover, the algorithm can be readily extended to the localization of networks with a one-hop transmission range distance measurement, and the propagation of the measurement error at one sensor node is limited within a small area of the network around the node. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing the superior performance of the proposed algorithm. © 2017 by the authors

Computational Conformal Geometry: A Review

Conformal geometry is considered as a fundamental topic in pure mathematics including complex analysis, algebraic geometry, Riemann surface theory, differential geometry and algebraic topology. Computational conformal geometry has an important role in digital geometry processing. A good number of practical algorithms are presented to compute conformal mapping, which has been broadly applied in a lot of practical fields such as computer graphics, wireless sensor networks, medical imaging, visualization, and so on.  This work reviews some major concepts and theorems of conformal geometry , their computational methods and the applications for surface parameterization