An alternative way to derive the contact rule and phase boundary rules for T-xiT-xi multicomponent phase diagrams

In order to predict the phase boundary dimension between two phase regions, Palatnik and Landau put forward the contact rule. Later, Zhao proposed phase boundary rules for T-xiT-xi multicomponent phase diagrams. In this study, the contact rule and phase boundary rules for T-xiT-xi multicomponent phase diagrams are derived from a geometric point of view, which contributes to a comprehensive understanding of the topological relations. The proposed formulas in this work differ slightly from those of Zhao, but can account for more geometric relations.