Abstract: To address collision avoidance and the intractable trajectory planning for unmanned aerial vehicles (UAVs) in narrow environments, this paper proposes exact collision avoidance formulations based on theories and methods from geometry, convex optimization, and numerical optimization, and then constructs the trajectory planning problem involving collision avoidance as an optimal control problem (OCP). The algorithm first represents the geometry of the UAV, obstacles, and the environment as a polyhedron, and with these representations, the collision avoidance requirements are transformed into mathematical problems concerning the separation between two sets and the containment of one set within another. Secondly, by using geometric methods and the hyperplane separation theorem, differentiable collision avoidance constraints are constructed in explicit ways. Subsequently, using the UAV kinematic model, the aforementioned OCP is discretized into a Nonlinear Programming (NLP) problem via the multiple shooting method, and then solved using the interior-point optimizer IPOPT. Finally, a collision avoidance scenario is constructed where a UAV must avoid obstacles within a narrow tube. Simulation results demonstrate that the constructed collision avoidance formulations are differentiable and smooth, and the proposed planning method can plan feasible and safe collision-free trajectories for UAVs in narrow environments. The results indicate that the proposed methods lay a technical foundation for exact collision avoidance and reliable trajectory planning in such environments.