Abstract: To solve the common problem of calculating maximum shape coefficient of the roots rotor and simplify the application problem of the existing zero curvature-radius method; based on the common two conjugate-contour constructions from the inside to the outside and from the outside to the inside; from analyzing the causality of engagement angle and shape coefficient, the contour coordinate equations with engagement angle as variable are derived; from analyzing the internal relation of contour interference and the interference coordinate, the analytic method with zero coordinate derivative is proposed for the minimum engagement angle; from analyzing the internal relation of the maximum drive angle and the limit phase angle, the analytic method with the maximum drive angle is proposed for the limit phase angle; finally, take the involute, arc and linear rotor as examples to demonstrate the innovative-method application in two contour constructs. All results show that the engagement angle and the shape coefficient are directly causal; the smaller the engagement angle, the larger the shape coefficient; the innovative method of the zero coordinate derivative plus the maximum drive angle is equivalent to the existing zero curvature-radius method, which can overcome the application problem of the existing zero curvature-radius method; the minimum engagement angle and limit phase angle are zero in the conjugate-contour construction from the inside to the outside, but in the conjugate-contour construction from the outside to the inside, the minimum engagement angle is not zero, the limit phase angle is determined by the zero drive angle derivative when the drive angle is not a monotonic function, and the end-point phase angle when the drive angle is a monotonic function, etc. It is concluded that the engagement angle replaces the shape coefficient as the contour construction variable, and the zero coordinate derivative plus the maximum drive angle replaces the zero-curvature radius, with clearer logic, simpler method and better analytics, which provides a theoretical basis for the curve-type innovation of the conjugate contour.