Optimal 3D Road Alignment on Topographic Surfaces: A Convergent Dynamic Programming Approach
We consider the problem of finding an optimal 3D road trajectory between two points on a terrain with variable elevation. Unlike common heuristic pathfinding methods, we propose a rigorous framework based on the calculus of variations, introducing an integral cost functional that incorporates material delivery and construction expenses. The existence of a global minimizer is established via the Arzelà--Ascoli theorem. To solve the problem numerically, we develop a dynamic programming scheme and provide a formal convergence proof. We prove that the sequence of piecewise-linear solutions converges to the true optimum when the grid discretization steps follow a specific power-law relation -- specifically, when the vertical step size decays faster than the horizontal one. To enhance efficiency, we introduce a local-search modification that reduces computational complexity to nearly quadratic O(τ^{-2-ε}), where τ is the discretization step along the x-axis. Numerical experiments on 2D and 3D terrains validate the theoretical results, showing that our approach achieves accuracy comparable to the Ritz method while significantly reducing processing time.