Abstract: The processing technology for round tube in five-axis CNC pipe cutting machine cannot completely support the cutting for square tube. Relative to the traditional intersecting model,a new model named 'Circumcircle Enveloping' based on round-square tube space intersecting mathematical model is developed to solve the longitudinal axis compensation. The mathematical models of square tube and related parameters at different intersecting sections,such as dihedral angle,groove angle,torch pose angle,were studied. A method named 'Adaptive Points' and speed control at inflexion positions are proposed to control the precision of cutting path. The experimental results verify the validity and applicability of the above models,which provides theoretical basis and ideas for developing multi-function CNC pipe cutting machines.