We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.

We study the computation of the straight skeleton of a planar straight-line graph (PSLG) by means of the triangulation-based wavefront propagation proposed by Aichholzer and Aurenhammer in 1998, and provide both theoretical and practical insights. As our main theoretical contribution we explain the algorithmic extensions and modifications of their algorithm necessary for computing the straight skeleton of a general PSLG within the entire plane, without relying on an implicit assumption of general position of the input, and when using a finiteprecision arithmetic. We implemented this extended algorithm in C and report on extensive experiments. Our main practical contribution is (1) strong experimental evidence that the number of flip events that occur in the kinetic triangulation of real-world data is linear in the number n of input vertices, (2) that our implementation, Surfer, runs in O(n log n) time on average, and (3) that it clearly is the fastest straight-skeleton code currently available. Introduction MotivationThe straight skeleton of a simple polygon is a skeletal structure similar to the generalized Voronoi diagram, but comprises straight-line segments only. It was introduced to computational geometry by Aichholzer et al. [1], and later generalized to planar straight-line graphs (PSLGs) by Aichholzer and Aurenhammer [2]. Currently, the most efficient straight-skeleton algorithm for PSLGs, by Eppstein and Erickson [7], has a worst-case time and space complexity of O(n 17 /11+ǫ ) for any ǫ > 0. For a certain class of simple polygons with holes Cheng and Vigneron [6] presented a randomized algorithm that runs in O(n √ n log 2 n) time.However, both algorithms employ elaborate data structures in order to achieve these complexities and are not suitable for implementation. The first comprehensive straight-skeleton code was implemented by Cacciola [4] and is shipped with the CGAL library [5]. It handles polygons with holes as input, and requires O(n 2 log n) time and O(n 2 ) space for real-world datasets, ⋆ The authors would like to thank Willi Mann for valuable discussions and comments.

We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams.

We introduce additively-weighted straight skeletons as a new generalization of straight skeletons. An additively-weighted straight skeleton is the result of a wavefront-propagation process where, unlike in previous variants of straight skeletons, wavefront edges do not necessarily start to move at the begin of the propagation process but at later points in time. We analyze the properties of additively-weighted straight skeletons and show how to compute straight skeletons with both additive and multiplicative weights.We then show how to use additively-weighted and multiplicatively-weighted straight skeletons to generate roofs and terrains for polygonal shapes such as the footprints of buildings or river networks. As a result, we get an automated generation of roofs and terrains where the individual facets have different inclinations and may start at different heights.

We study the practical computation of mitered and beveled offset curves of planar straight-line graphs (PSLGs), i.e., of arbitrary collections of straight-line segments in the plane that do not intersect except possibly at common end points. The line segments can, but need not, form polygons. Similar to Voronoi-based offsetting, we propose to compute a straight skeleton of the input PSLG as a preprocessing step for mitered offsetting. For this purpose, we extend and adapt Aichholzer and Aurenhammer's triangulation-based straight-skeleton algorithm to make it process real-world data on a conventional finite-precision arithmetic.We implemented this extended algorithm in C and use our implementation for extensive experiments. All tests demonstrate the practical suitability of using straight skeletons for the offsetting of complex PSLGs. Our main practical contribution is strong experimental evidence that mitered offsets of PSLGs with 100 000 segments can be computed in about ten milliseconds on a standard PC once the straight skeleton is available and that our implementation clearly is the fastest code for mitered offsetting even if the computational costs of the straight-skeleton computation are included in the timings.