Discrete Geometric Path Analysis in Computational Geometry

The geometric shortest path problem is one of the fundamental problems in Computational Geometry and related fields.In the first part of this thesis, we study the weighted region problem (WRP), which is to compute a geometric shortest path on a weighted partitioning of a plane.Recent results show that WRP is not solvable in any algebraic computation model over rational numbers.Thus, scientists have focused on approximate solutions.We first study the WRP when the input partitioning of space is an arrangement of lines.We provide a technique that makes it possible to apply the existing approximation algorithms for triangulations to arrangements of lines.Then, we formulate two qualitative criteria for weighted short paths.We show how to produce a path that is quantitatively close-to-optimal and qualitatively satisfactory.The results of our experiments carried out on triangular irregular networks (TINs) show that the proposed algorithm could save, on average, 51% in query time and 69% in memory usage, in comparison with the existing method.In the second part of the thesis, we study some variants of the Fréchet distance.The Fréchet distance is a well-studied and commonly used measure to capture the similarity of polygonal curves.All of the problems that we studied here can be reduced to a geometric shortest path problem in configuration space.Firstly, we study a robust variant of the Fréchet distance since the standard Fréchet distance exhibits a high sensitivity to the presence of outliers.Secondly, we propose a new measure to capture similarity between polygonal curves, called the minimum backward Fréchet distance (MBFD).More specifically, for a given threshold ε, we are searching for a pair of walks for two entities on the two input polygonal curves such that the union of the portions of required backward movements is minimized and the distance between the two entities, at any time during the walk, is less than or equal to ε. Thirdly, we generalize MBFD to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous.More specifically, each edge of input polygonal curves has an associated non-negative weight.The cost of backtracking on an edge is the Euclidean length of backward movement on that edge multiplied by the corresponding edge weight.Lastly, for a given graph H, a polygonal curve T , and a threshold ε, we propose a geometric algorithm that computes a path, P , in H, and a parameterization of T , that minimize the sum of the length of walks on T and P whereby the distance between the entities moving along P and T is at most ε, at any time during the walks.iv Prima facie, I am grateful to the God for well-being that were necessary to complete this thesis.

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