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This paper presents several results on some cost-minimizing path problems in polygonal regions. For these types of problems, an approach often used to compute approximate optimal paths is to apply a discrete search algorithm to a graph Gepsi constructed from a discretization of the problem; this graph is guaranteed to contain an epsi-good approximate optimal path, i.e., a path with a cost within (1 + epsi) factor of that of an optimal path, between given source and destination points. Here, epsi > 0 is the user-defined error tolerance ratio. We introduce a class of piecewise pseudo-Euclidean optimal path problems that includes several non-Euclidean optimal path problems previously studied and show that the BUSHWHACK algorithm, which was formerly designed for the weighted region optimal path problem, can be generalized to solve any optimal path problem of this class. We also introduce an empirical method called the adaptive discretization method that improves the performance of the approximation algorithms by placing discretization points densely only in areas that may contain optimal paths. It proceeds in multiple iterations, and in each iteration, it varies the approximation parameters and fine tunes the discretization.