Gradient-based optimisation is a common approach employed by local planners to rapidly find efficient and feasible trajectories for autonomous UAVs. Notably, B-spline or Bézier trajectory parameterisations decrease computation times while allowing for the continual enforcement of safety and kinodynamic feasibility constraints. However, current B-spline- or Bézier-based techniques that enforce collision constraints remain computationally expensive, whilst employing a conservative spline envelope. Conversely, methods that focus on minimising computational cost do not explicitly enforce a collision constraint, relying instead on arbitrary tuning parameters with unconstrained algorithms, and resorting to re-planning when the optimisation produces a colliding or dynamically infeasible trajectory. In this paper, a novel, tight, computationally efficient envelope that bounds a B-spline trajectory with respect to its control points is introduced, enabling the formulation of a constrained optimisation problem that converges rapidly while ensuring the safety and feasibility of the generated trajectories. The proposed algorithm is tested using an open-source simulation of a real-world UAV, demonstrating that, although each optimisation takes longer than unconstrained techniques, the method is still sufficiently fast for real-time applications. Furthermore, the B-spline envelope ensures that the resulting trajectories are less conservative than existing approaches, offering improved operational efficiency.