Results on geometric networks and data structures

This thesis discusses four problems in computational geometry. In traditional colored range-searching problems, one wants to store a set of n objects with m distinct colors for the following queries: report all colors such that there is at least one object of that color intersecting the query range. Such an object, however, could be an `outlier' in its color class. We consider a variant of this problem where one has to report only those colors such that at least a fraction t of the objects of that color intersects the query range, for some parameter t. Our main results are on an approximate version of this problem, where we are also allowed to report those colors for which a fraction (1-epsilon)t intersects the query range, for some fixed epsilon > 0. We present efficient data structures for such queries with orthogonal query ranges in sets of colored points, and for point stabbing queries in sets of colored rectangles. A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. R-trees are box-trees with nodes of high degree. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. The geometric minimum-diameter spanning tree (MDST) of a set of n points is a tree that spans the set and minimizes the Euclidian length of the longest path in the tree. So far, the MDST can only be found in slightly subcubic time. We give two fast approximation schemes for the MDST, i.e. factor-(1+epsilon) approximation algorithms. One algorithm uses a grid and takes time O*(1/epsilon^(5 2/3) + n), where the O*-notation hides terms of type O(log^O(1) 1/epsilon). The other uses the well-separated pair decomposition and takes O(1/epsilon^3 n + (1/epsilon) n log n) time. A combination of the two approaches runs in O*(1/epsilon^5 + n) time. The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the L1-distance. We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain the following results: for arbitrary graphs, the problem is NP-hard; for trees, we can solve the problem by linear programming on O(n^2) variables and constraints; for paths, we can solve the problem in time O(n^3 log n); for rectangles sorted vertically along a path, the problem can be solved in O(n^2) time

Implicit flow routing on terrains with applications to surface networks and drainage structures

Flow-related structures on terrains are defined in terms of paths of steepest descent (or ascent). A steepest descent path on a polyhedral terrain T with n vertices can have T(n^2) complexity. The watershed of a point p --- the set of points on T whose paths of steepest descent reach p --- can have complexity T(n^3). We present a technique for tracing a collection of n paths of steepest descent on T implicitly in O(n logn) time. We then derive O(n log n) time algorithms for: (i) computing for each local minimum p of T the triangles contained in the watershed of p and (ii) computing the surface network graph of T. We also present an O(n^2) time algorithm that computes the watershed area for each local minimum of T

On IO-efficient viewshed algorithms and their accuracy

Given a terrain T and a point v, the viewshed or visibility map of v is the set of points in T that are visible from v. To decide whether a point p is visible one needs to interpolate the elevation of the terrain along the line-of-sight (LOS) vp. Existing viewshed algorithms differ widely in which and how many points they chose to interpolate, how many lines-of-sight they consider, and how they interpolate the terrain. These choices crucially affect the running time and accuracy of the algorithms. In this paper our goal was to obtain an IO-efficient algorithm that computes the viewshed on a grid terrain with as much accuracy as possible given the resolution of the data. We describe two algorithms which are based on computing and merging horizons, and we prove that the complexity of horizons on a grid of n points is O(n), improving on the general O(na(n)) bound on triangulated terrains. Our finding is that, in practice, horizons on grids are significantly smaller than their theoretical worst case bound, which makes horizon-based approaches very fast. To measure the differences between viewsheds computed with various algorithms we implement an error metric that averages differences over a large number of viewsheds computed from a set of viewpoints with topological significance, like valleys and ridges. Using this metric we compare our current approach, Van Kreveld's model used in our previous work [7], the algorithm of Ferreira et al. [6], and the viewshed module r.los in the open source GIS GRASS

Topologically safe curved schematization

Traditionally schematized maps make extensive use of curves. However, automated methods for schematization are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematization that allows a choice of quality measures and curve types. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs

Kennisbasisthema: Ketens en Agrologistiek

Wageningen UR voert een meerjarig onderzoeksprogramma uit, getiteld Ketens en Agrologistiek. Deze projectbundel geeft informatie over projecten in dit thema en van elk project is een flyer opgenome

Polarized emission from hexagonal-silicon-germanium nanowires

We present polarized emission from single hexagonal silicon-germanium (hex-SiGe) nanowires. To understand the nature of the band-to-band emission of hex-SiGe, we have performed photoluminescence spectroscopy to investigate the polarization properties of hex-SiGe core-shell nanowires. We observe a degree of polarization of 0.2 to 0.32 perpendicular to the nanowire c-axis. Finite-difference time-domain simulations were performed to investigate the influence of the dielectric contrast of nanowire structures. We find that the dielectric contrast significantly reduces the observable degree of polarization. Taking into account this reduction, the experimental data are in good agreement with polarized dipole emission perpendicular to the c-axis, as expected for the fundamental band-to-band transition, the lowest energy direct band-to-band transition in the hex-SiGe band structure.</p

The sound of space-filling curves

This paper presents an approach for representing space-filling curves by sound, aiming to add a new way of perceiving their beautiful properties. In contrast to previous approaches, the representation is such that geometric similarity transformations between parts of the curve carry over to auditory similarity transformations between parts of the sound track. This allows us to sonify space-filling curves, in some cases in up to at least five dimensions, in such a way that some of their geometric properties can be heard. The results direct attention to the question whether space-filling curves exhibit a structure that is similar to music. I show how previous findings on the power spectrum of pitch fluctuations in music suggest that the answer depends on the number of dimensions of the space-filling curve

How many three-dimensional Hilbert curves are there?

Hilbert's two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher dimensions. In fact, the number of ways in which this may be done ranges from zero to infinite, depending on what properties of the Hilbert curve one considers to be essential. In this work we take the point of view that a Hilbert curve should at least be self-similar and traverse cubes octant by octant. We organize and explore the space of possible three-dimensional Hilbert curves and the potentially useful properties which they may have. We discuss a notation system that allows us to distinguish the curves from one another and enumerate them. This system has been implemented in a software prototype, available from the author's website. Several examples of possible three-dimensional Hilbert curves are presented, including a curve that visits the points on most sides of the unit cube in the order of the two-dimensional Hilbert curve; curves of which not only the eight octants are similar to each other, but also the four quarters; a curve with excellent locality-preserving properties and endpoints that are not vertices of the cube; a curve in which all but two octants are each other's images with respect to reflections in axis-parallel planes; and curves that can be sketched on a grid without using vertical line segments. In addition, we discuss several four-dimensional Hilbert curves