In this paper, we examine a unit quaternion based method to design the optimal paths with maximum sun exposure for unmanned aerial vehicles (UAVs) equipped with photovoltaic cells on their wings. The mission of traveling between two specified boundary points with fixed flying time and constant speed is considered. Since the solar power is the sole source of energy for these UAVs during the flight, we consider the problem of maximizing the incoming solar radiation throughout their trajectory. As the attitude of the UAV directly determines solar intensity normal to the vertical surface of the wing, we use a unit quaternion based method to control the attitude maneuver during the flight interval. Subsequently, the aircraft kinematics are expressed as quadratic functions in terms of unit quaternions which can be solved by a branch and bound approach. Simulation results in two and three dimensions are presented.