We consider two algorithms for obtaining a desired point on the central-path as identified by a fixed positive real number. The first and more general algorithm is based on Nesterov and Nemirovskii's long-step method. Using their analysis, we are able to provide polynomial complexity bounds in a straightforward manner. The second algorithm is suitable for conic problems involving self-scaled cones. This algorithm is less general but more efficient and polynomial complexity bounds are observed using existing complexity results. Both algorithms are suitable for, but not restricted to, solving a class of problems associated with recentred barrier function model predictive control.