Morphing of Building Footprints Using a Turning Angle Function

We study the problem of morphing two polygons of building footprints at two different scales. This problem frequently occurs during the continuous zooming of interactive maps. The ground plan of a building footprint on a map has orthogonal characteristics, but traditional morphing methods cannot preserve these geographic characteristics at intermediate scales. We attempt to address this issue by presenting a turning angle function-based morphing model (TAFBM) that can generate polygons at an intermediate scale with an identical turning angle for each side. Thus, the orthogonal characteristics can be preserved during the entire interpolation. A case study demonstrates that the model yields good results when applied to data from a building map at various scales. During the continuous generalization, the orthogonal characteristics and their relationships with the spatial direction and topology are well preserve

Applied Similarity Problems Using Frechet Distance

In the first part of this thesis, we consider an instance of Frechet distance problem in which the speed of traversal along each segment of the curves is restricted to be within a specfied range. This setting is more realistic than the classical Frechet distance setting, specially in GIS applications. We also study this problem in the setting where the polygonal curves are inside a simple polygon. In the second part of this thesis, we present a data structure, called the free-space map, that enables us to solve several variants of the Frechet distance problem efficiently. Our data structure encapsulates all the information available in the free-space diagram, yet it is capable of answering more general type of queries efficiently. Given that the free-space map has the same size and construction time as the standard free-space diagram, it can be viewed as a powerful alternative to it. As part of the results in Part II of the thesis, we exploit the free-space map to improve the long-standing bound for computing the partial Frechet distance and obtain improved algorithms for computing the Frechet distance between two closed curves, and the so-called minimum/maximum walk problem. We also improve the map matching algorithm for the case when the map is a directed acyclic graph. As the last part of this thesis, given a point set S and a polygonal curve P in R^d, we study the problem of finding a polygonal curve Q through S, which has a minimum Frechet distance to P. Furthermore, if the problem requires that curve Q visits every point in S, we show it is NP-complete.Comment: arXiv admin note: text overlap with arXiv:1003.0460 by other author

Straight Line Movement in Morphing and Pursuit Evasion

Piece-wise linear structures are widely used to define problems and to represent simplified solutions in computational geometry. A piece-wise linear structure consists of straight-line or linear pieces connected together in a continuous geometric environment like 2D or 3D Euclidean spaces. In this thesis two different problems both with the approach of finding piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion and straight-line morphing. Straight-line pursuit evasion is a geometric version of the famous cops and robbers game that is defined in this thesis for the first time. The game is played in a simply connected region in 2D. It is a full information game where the players take turns. The cop’s goal is to catch the robber. In a turn, each player may move any distance along a straight line as long as the line segment connecting their current location to the new location is not blocked by the region’s boundary. We first prove that the cop can always win the game when the players move on the visibility graph of a simple polygon. We prove this by showing that the visibility graph of a simple polygon is “dismantlable” (the known class of cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other settings of the game are also studied in this thesis: when the players are free to move on the infinitely many points inside a simple polygon, and inside a splinegon. In both cases we show that the cop can always win the game. For the case of polygons, the proposed cop strategy gives an asymptotically tight linear bound on the number of steps the cop needs to catch the robber. For the case of splinegons, the cop may need a quadratic number of steps with the proposed strategy, while our best lower bound is linear. Straight-line morphing is a type of morphing first defined in this thesis that provides a nice and smooth transformation between straight-line graph drawings in 2D. In straight- line morphing, each vertex of the graph moves forward along the line segment connecting its initial position to its final position. The vertex trajectories in straight-line morphing are very simple, but because the speed of each vertex may vary, straight-line morphs are more general than the commonly used “linear morphs” where each vertex moves at uniform speed. We explore the problem of whether an initial planar straight-line drawing of a graph can be morphed to a final straight-line drawing of the graph using a straight-line morph that preserves planarity at all times. We prove that this problem is NP-hard even for the special case where the graph drawing consists of disjoint segments. We then look at some restricted versions of the straight-line morphing: when only one vertex moves at a time, when the vertices move one by one to their final positions uninterruptedly, and when the edges morph one by one to their final configurations in the case of disjoint segments. Some of the variations are shown to be still NP-complete while some others are solvable in polynomial time. We conjecture that the class of planar straight-line morphs is as powerful as the class of planar piece-wise linear straight-line morphs. We also explore a simpler problem where for each edge the quadrilateral formed by its initial and final positions together with the trajectories of its two vertices is convex. There is a necessary condition for this case that we conjecture is also sufficient for paths and cycles

Morphing Parallel Graph Drawings

A pair of straight-line drawings of a graph is called parallel if, for every edge of the graph, the line segment that represents the edge in one drawing is parallel with the line segment that represents the edge in the other drawing. We study the problem of morphing between pairs of parallel planar drawings of a graph, keeping all intermediate drawings planar and parallel with the source and target drawings. We call such a morph a parallel morph. Parallel morphs have application to graph visualization. The problem of deciding whether two parallel drawings in the plane admit a parallel morph turns out to be NP-hard in general. However, for some restricted classes of graphs and drawings, we can efficiently decide parallel morphability. Our main positive result is that every pair of parallel simple orthogonal drawings in the plane admits a parallel morph. We give an efficient algorithm that computes such a morph. The number of steps required in a morph produced by our algorithm is linear in the complexity of the graph, where a step involves moving each vertex along a straight line at constant speed. We prove that this upper bound on the number of steps is within a constant factor of the worst-case lower bound. We explore the related problem of computing a parallel morph where edges are required to change length monotonically, i.e. to be either non-increasing or non-decreasing in length. Although parallel orthogonally-convex polygons always admit a monotone parallel morph, deciding morphability under these constraints is NP-hard, even for orthogonal polygons. We also begin a study of parallel morphing in higher dimensions. Parallel drawings of trees in any dimension always admit a parallel morph. This is not so for parallel drawings of cycles in 3-space, even if orthogonal. Similarly, not all pairs of parallel orthogonal polyhedra admit a parallel morph, even if they are topological spheres. In fact, deciding parallel morphability turns out to be PSPACE-hard for both parallel orthogonal polyhedra, and parallel orthogonal drawings in 3-space