geometric modeling of dubins airplane movement and its metric

the time-optimal trajectory for an airplane from some starting point to some final point is studied by many authors. here, we consider the extension of robot planer motion of dubins model in three dimensional spaces. in this model, the system has independent bounded control over both the altitude velocity and the turning rate of airplane movement in a non-obstacle space. here, in this paper a geometrization of time-optimal trajectory of dubins airplane is also obtained. more intuitively, the metric related to this phenomenon is described as a geometry in space. it is shown that the distance traveled in movement of an airplane obeys certain conditions of a well-known geometry called finsler geometry. moreover, it is proved that the geometry of movement of an airplane is a special finsler metric known as randers metric, and therefore, time-optimal paths are geodesics of randers metric.