Based on the mathematical geometry theory

the basic profile equations of the female-male rotor were derived through coordinate transformation and envelope conditions

and the mathematical model of the double-tooth gas circulation pump was established.The relationship between the position of the meshing point and the rotor angle of the female rotor with the meshing of the male and female rotors was studied.The calculation method of line integral was used to derive the relationship between the elementary area of the working volume and the rotor angle

which can simplify the calculation difficulty of the working volume.Results showed that when the rotor angles were 0 and π

the female and male rotors formed a closed invalid area

which led to the sudden rise and fall of the element area of the working cavity during the change of the rotor angle.At the end of inhalation

the suction chamber had an additional part connected to the suction port compared with the exhaust chamber

resulting in a sudden drop in the element area.The establishment of these basic geometric theories can lay a certain theoretical foundation for the application of double-tooth gas circulation pumps.