The shortest path problem in continuous 2D spaces with polygonal obstacles is a fundamental challenge in domains ranging from robotics to ship navigation. While visibility graphs provide exact solutions, they become prohibitively large for massive datasets, such as a high-resolution world map, potentially exceeding a terabyte in size. This thesis investigates the θ-spanner as a memory-efficient approximation of the visibility graph, allowing for a configurable trade-off between graph size and path accuracy. We perform a comprehensive empirical analysis of θ-spanners, demonstrating that the number of cones required to achieve low approximation errors in practice is significantly lower than theoretical worst-case bounds suggest. When compared to the state-ofthe- art Constrained Delaunay Triangulation (CDT) approach, θ-spanners are shown to provide a competitive, sparse representation that yields accurate paths without requiring post-processing algorithms. Furthermore, this work introduces two novel modifications: First, we propose an optimized θ-Spanner, which selectively adds edges with high approximation errors to ensure a strict upper bound on path deviation. Second, we present the θ-augmented Shrunk Graph, a method that utilizes spanner construction to recover vertices in reduced graphs. This approach generates lower-bound distance approximations that are tighter—approximately 3.43 times closer to the ground truth on average—than previous triangle-based lower-bound methods. AI was used to improve spelling and readability.