To solve the problem of lightweight roots rotor volume utilization maximization,an extreme contour construction method was proposed in which the inner conjugate contour segment was converged to a fixed-point.According to the conjugate geometry of the extreme contour,the extreme contour coefficient and the extreme volume utilization coefficient of the two types of rotors were calculated;a double-symmetry method was used to derive the construction equation for the contour segment to be obtained.The results show that the fixed-point conjugating contour has an extreme shape factor and volume utilization factor;by using the high-shaped construction method,the better lightweight effect can be further obtained and multiple sealing-points for relieving conjugate leakage in gas medium was formed;through the substitute arc inside pitch circle,the trapped-oil phenomenon in the liquid medium can be effectively eliminated,the contour is simpler and the processing is easier.