A jack-up is an off-shore platform that, thanks to the capability of lowering its legs, is capable to elevate its hull from the water. Typically, the movement of the leg and the subsequent emersion of the jack-up is ruled by a rack-pinion system in which the pinion is a low number of teeth (typically 7 ÷ 8) spur gear. Recently, in order to reduce the contact pressure, it has been proposed to substitute the classical Involute (INV) profile of the pinion by a Multi Radius (MR) one. However, due to their intrinsic differences, the classical INV theory cannot be adopted directly as means to study the behaviour of the MR profile. Therefore, a Tooth Contact Analysis (TCA) procedure for the initial evaluation of a rack and MR pinion system is presented here. The aforementioned TCA enables to calculate relevant gear quantities (e.g. forces, velocities and contact pressure). Furthermore, thanks to the concept of coupled Transmission Error functions, it allows to evaluate, in the initial design phase, if the rack and pinion mesh occurs without impacts.