Let $\Pi$ be a link projection in $S^2$. John Conway [5] and later Francis Bonahon and Larry Siebenmann [2] undertook to split $\Pi$ into canonical pieces. These pieces received different names: basic or polyhedral diagrams on one hand, rational, algebraic, bretzel, arborescent diagrams on the other hand. This paper proposes a thorough presentation of the theory, known to happy fews. We apply the existence and uniqueness theorem for the canonical decomposition to the classification of Haseman circles and to the localisation of the flypes.