Contact dynamics relies on the simultaneous satisfaction of constraints at the robot body level and at the contact level. At both levels, various formulations can be chosen that all must lead to the same results, given the same hypothesis, hence the little importance of their details. Yet when using it in an optimal control problem, a particular formulation is often imposed by the task to be performed by the robot. In this paper, we detail the formulation of the contact quantities (force, movement) in an arbitrary frame imposed by the task. In that case, we will show that we are typically not interested in working in the local frame (attached to the robot contact point), nor in the world frame, but in a user-defined frame centered at the contact location with a fixed orientation in the world. The derivations can then be used for 6D, 3D or normal (pure-sliding) contact. We implemented the corresponding derivatives on top of the contact dynamics of Pinocchio in the optimal control solver Crocoddyl. We show that a unique formulation is able to handle several operational orientations by achieving several surfacing tasks in model predictive control with the robot Talos.