This paper addresses a UAV path planning task that seeks to observe a set of objects while satisfying the observation quality constraint. A dynamic programming algorithm is proposed that enables the UAV to observe the target objects with the shortest path while subjecting to the observation quality constraint. The objects have their own facing direction and restricted observation range. With an observing order, the algorithm achieves $(1+ε)$-approximation ratio in theory and runs in polynomial time. The extensive results demonstrate that the algorithm produces near-optimal solutions, the effectiveness of which is also tested and proved in the Airsim simulator, a realistic virtual environment.