We present an algorithm for computing exact shortest paths, and consequently distances, from a generalized source (point, segment, polygonal chain or polygonal region) on a polyhedral terrain in which polygonal chain or polygon obstacles are allowed. We also present algorithms for computing discrete Voronoi diagrams of a set of generalized sites (points, segments, polygonal chains or polygons) ...

We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order-j Voronoi diagrams of m sites, for j = 1, . . . , k, is O(kn + km+ knm), which is asymptotically tight...

Cellular structures manifest their outstanding mechanical properties in many biological systems. One key challenge for designing and optimizing these geometrically complicated lies devising an effective geometric representation to characterize the system's spatially varying cellular evolution driven by objective sensitivities. A conventional discrete structure, e.g., a Voronoi diagram, whose re...

Voronoi diagrams are an important data structure in computer science. However well studied mathematically, understanding such diagrams for different metrics, orders, and site shapes is a complex task. We propose a new method to visualize k-order diagrams and give an efficient adaptive implementation for this method. The algorithm is easy to customize for different metrics and site shapes. Its r...