The shortest path problem is a well-studied problem in computer-science. For transport networks, there exist natural graph representations and highly efficient algorithms that can compute shortest paths on millions of nodes within milliseconds. In contrast, computing shortest paths in space (e.g. in R^2) poses some challenges. Shortest path computations in space have applications in robotics, naval routing or video games. As shortest paths in a continuum are hard to compute (with the Euclidean shortest-path problem in 3D even proven to be NP-hard), approximations can be necessary to obtain acceptable runtimes. In this thesis, an approximation scheme is studied that guarantees solutions with cost at most (1+Īµ) times the optimum. It uses a triangulation of the domain and, given Īµ, construct a discretization. By performing a Dijkstra search, one can then approximate shortest paths with the given quality guarantee. This scheme is implemented and its practicality evaluated on larger instances.