On Computing Straight Skeletons by Means of Kinetic Triangulations

Abstract

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.

Computing Mitered Offset Curves Based on Straight Skeletons

“…If the set of all triangles involved in the multi-flip event cloud of the second setting could be assumed to be correct -for instance because the algorithm is run with exact arithmetic -then one can devise an ordering that ensures their eventual resolution [20]. However, with limited precision even establishing that set is impossible.…”

“…In Palfrader et al [21], we sketched the theoretical basis of an extension and modification 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.…”

“…More recently, while implementing our straightskeleton code Surfer based on the theoretical basis laid out in [21], we investigated the peculiarities of a realization of that algorithm on a standard floatingpoint arithmetic. For instance, we refine the naive (determinant-based) determination of the collapse times in order to improve the numerical reliability of the algorithm.…”

Generalized offsetting of planar structures using skeletons

“…The straight skeleton tessellates the input polygon into non-overlapping faces. We refer to Huber [13] for a detailed discussion of straight skeletons; recent algorithms and codes by the authors for computing straight skeletons are documented in [14,17].…”

Route Planning in 3d Environment with a Multivariant Model

“…Рельеф поверхности функции на графическом образе (рисунок 9в) формирует скелет общего контура описания сцены, широко применимый при анализе его формы и ко многим другим прикладным задачам [28,29].…”