The Euclidean shortest path problem is a problem in computational geometry, which requires finding a shortest path between two points in the Euclidean plane avoiding a set of polygonal obstacles. This thesis explores the analog of this problem on a surface with different geometry, the hyperbolic plane. To remain within the Euclidean framework, we introduce the Beltrami-Klein and Poincar´e disk models - two prominent map projections of the hyperbolic plane into the Euclidean plane. Using these models, we adapt and implement the canonical approach that solves the Euclidean shortest path problem for the hyperbolic plane. This approach involves triangulating the polygonal domain to derive the visibility graph from the set of obstacles. A query between two points can then be answered by applying shortest-path algorithms, such as A* or Dijkstra, on this graph. Furthermore, we evaluate an approximation scheme that runs these shortest-path algorithms on a subgraph of the triangulation. Due to the absence of polygonal land maps in the hyperbolic plane, an algorithm to generate them has been developed and implemented.