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In this article, the optimal design of a road joining two terminals is investigated. A geometric model is proposed including horizontal transition curves and vertical curves, obtaining parameterizations for the central axis of the road as well as for its entire surface. These parameterizations allow to express and compute, with great simplicity, the major infrastructure costs, including land acquisition, clearance, pavement, maintenance, and earthwork, where multiple layers of materials with different costs can be handled. The road design problem is formulated as a smooth constrained optimization problem and a two-stage algorithm is suggested for its numerical resolution. Finally, numerical results are presented in an academic test and in a case study that propose designing a bypass in a Spanish national road (N-640) to avoid crossing Monterroso's town center.

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