Inverse optimal control (IOC) is a framework used in many fields, especially in robotics and human motion analysis. In this context, various methods of resolution have been proposed in the literature. This article presents Projected Inverse Optimal Control (PIOC), an approach that offers a simple and comprehensive view of IOC methods. Especially, we explain how uncertainties can be properly addressed in our view. Thus, this article highlights how classical methods can be understood as projections of trajectories in the solution space of the underlying Direct Optimal Control (DOC) problem. This perspective allows for an examination of projections other than the classical methods, which can be fruitful for researchers in the field. As an example, we propose a projection that allows us to choose the underlying cost functions of an IOC problem from a set. The IOC's sub-problems are also addressed, such as modelling observed trajectories, noise measurement and the reliability of solutions obtained by IOC. Our proposal is supported by a simple and canonical example throughout the document.