This dissertation explores the problem of transmission line (TL) routing through finding the shortest path on an undirected graph with no improving cycles, considering quadratic costs for adjacent arcs. This problem is known as the Adjacent Quadratic Shortest Path Problem (AQSPP). This work provides the theoretical background for the AQSPP, proposes an extension of Dijkstra s algorithm (aqDijkstra) for solving AQSPP in polynomial-time and discusses how AQSPP can be included in routing methodologies. Furthermore, it is presented an improvement to the algorithm: the adjacent quadratic A star (aq A star) with a backward search for cost-togo estimation, to speed up search. For computational experiments, aqDijkstra and aqA star are benchmarked with other algorithms from the technical literature. The search behavior of the algorithms is also studied within different tests, including: quadratic cost variation, randomly generated graph instances and increasingly larger instances. The numerical results suggests that: (i) aqA star outperformed all the other algorithms, being 40 times faster than aqDijsktra and 50 times faster than the fastest benchmark algorithm; (ii) the studied algorithms do not lose efficiency as quadratic costs increase; (iii) aqA star and aqDijkstra were faster benchmark algorithms under random graph instances, indicating their robustness. Two applications are provided, one for illustrative purposes, and another to study performance on a real application. The aqA star algorithm solved an AQSSP on a graph with almost a billion quadratic arcs and provided a route with three times lower additional costs.