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Optimal turning climb-out and descent flight-paths from and to runway headings are derived to provide the missing elements of a complete flight-path optimization for minimum fuel consumption. The paths are derived by generating a field of extremals, using the necessary conditions of optimal control. Results show that the speed profiles for straight and turning flight are essentially identical, except for the final horizontal accelerating or decelerating turn. The optimal turns, which require no abrupt maneuvers, could easily be integrated with present climb-cruise-descent fuel-optimization algorithms.