Abstract: The static stability boundary problem of AC transmission line remains unresolved. The load safety domain can constrain the line voltage difference and maintain the safety level of the multi-level grid voltage, but the stability margin is unknown. Traditional static power-angle stability and voltage stability are different from the concept, mechanism and analysis method. It is found that a unified concept and mechanism can be formed for the two types of static stability of the line, and the sufficient condition for static stability of the whole grid is that each branch satisfies static stability. Based on the complex variable state equation of the line, the independent equations of complex power-voltage (P_j-Q_j-U_j) and complex power-power angle (P_j-Q_j-θ_ij) at the end are obtained. Through solvable condition analysis and partial derivation proof, the unique P_j-Q_j stability boundary and its characteristics are obtained, and the homogeneity of the two kinds of destabilize is proved. Based on analytical geometry, the stability boundaries and characteristics of U_j and θ_ij are derived to get a destabilizing shape interpretation that conforms to the actual situation. Based on the P_j-Q_j stability boundary, the stability margin index of the line load security domain and the correction method of the voltage safety level are proposed. The feasibility of the line model based on the head-end reference frame is theoretically verified, and the stability domain matching method of the upper and lower topology of the power grid is proposed. Finally, based on the state observability criterion, the independence of equations and the principle of equilibrium point generation, the limitation of line power-angle characteristic is analyzed. The examples show that the proposed stability boundary and stability domain matching methods are consistent with the convergence results of the PSASP program. Under the background that the large-scale access of new energy leads to the large fluctuation of line power flow, this research has more theoretical value and practical significance.