Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region

A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

Given a graph $G$ cellularly embedded on a surface $\Sigma$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $\Sigma$ along $G$ yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any $\varepsilon >0$, we show how to compute a $(1+ \varepsilon)$ approximation of the shortest cut graph in time $f(\varepsilon, g)n^3$. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which may be of independent interest

Three-dimensional foot shape analysis in children : a pilot analysis using three-dimensional shape descriptors

Existing clinical measures to describe foot morphology are limited in that they are commonly two-dimensional, low in resolution and accuracy, and do not accurately represent the multi-planar and complex changes during development across childhood. Using three-dimensional (3D) scanner technology provides the opportunity to understand more about morphological changes throughout childhood with higher resolution and potentially more relevant 3D shape measures. This is important to advance the prevailing arguments about the typical development of children's feet and inform the development of appropriate clinical measures. 3D shape descriptors derived from 3D scanning can be used to quantify changes in shape at each point of the 3D surface. The aim of this study was to determine whether 3D shape descriptors derived from 3D scanning data can identify differences in foot morphology between children of different ages. Fifteen children were recruited from three age groups (2, 5, and 7 years of age). Both feet were scanned in bipedal stance, using the Artec Eva (Artec Group, Luxembourg, Luxembourg) hand-held scanner. Three dimensional shape descriptors were extracted from the 3D scans of the right foot, to create histograms for each age group and heat maps of representative participants for comparison. There were changes to the dorsal, medial and lateral surfaces of the feet with age. The surfaces became less round along with an increase in indented areas. This is supported by the heat maps which demonstrated that the surfaces of the anatomical landmarks (e.g. the malleoli and navicular tuberosity) became more rounded and protruding, with indented surfaces appearing around these landmarks. On the plantar surface, the concavity of the midfoot was evident and this concavity extended into the midfoot from the medial aspect as age increased. The findings of this study indicated that with increasing age the foot becomes thinner in 3D, with bony architecture emerging, and the medial longitudinal arch (MLA) increases in area and concavity. Three-dimensional shape descriptors have shown good potential for locating and quantifying changes in foot structure across childhood. Three-dimensional shape descriptor data will be beneficial for understanding more about foot development and quantifying changes over time