This paper investigates, for the first time, a three-dimensional mathematical model for constructing an optimal trajectory that had previously been studied only in the two-dimensional setting. An integral cost functional of the trajectory is formulated and a necessary condition for its minimum is derived. The resulting integro-differential equation is solved by the Galerkin and collocation methods. A genetic algorithm is also proposed. Classical methods based on necessary extremum conditions are well suited for finding local optima in smooth problems, whereas the genetic algorithm is advantageous when exact methods are unavailable: it can escape local extrema and often produces solutions close to the global optimum. Results of numerical experiments are presented, and future research directions aimed at improving numerical schemes and developing hybrid approaches are outlined.