Many route-location problems can be regarded as the minimisation of some accumulated cost, impact, or similar friction per unit length. Analogies can be found with paths of light and other least-time paths, and with geodesics. The problem is often solved by finding a path through a lattice of sample costs, with the use of modified shortest-path algorithms. Lattice paths do not converge to continuous-space paths. The differences are shown to depend on the set of permitted moves in the lattice, with the use of three cases. The continuous-space problem is solved for surfaces described by simple functions and for choropleth surfaces, and compared with lattice solutions. Three heuristic approaches for large problems are reviewed, with emphasis placed on regular spatial aggregation.