## 1 Introduction
[PERSON] [20] presented a method for machining face gears with spherical hobs and verified its feasibility through experiments, but its machining efficiency is relatively low. [PERSON] [21] proposed that the tooth surface of the face gear is a modified and variable tooth thickness rack, but no in-depth research was conducted. [PERSON] [22] proposed a Coniflex tool for line contact machining of bevel gears, but the machined tooth surface deviates significantly from the theoretical tooth surface. [PERSON] and [PERSON] [23] proposed a new bevel gear tooth surface, which reduces the processing time by 10 times compared to traditional bevel gear tooth surfaces, and its matching opinions can also be easily processed on a five-axis machining center. According to the above research, it was found that due to the complex spatial surface of the face gear, it is necessary to limit the tool-to-point contact with the workpiece during generative machining. However, this idea can lead to high cost or low efficiency in face gear machining. [PERSON] [24] proposed that if the tooth surface of a face gear is directly regarded as a ruled surface, that is, a regular surface formed by a cluster of straight lines, its machining efficiency will be improved, and the machining cost will be reduced, which is conducive to the promotion and application of face gears. [PERSON] [25] studied a simplified mathematical model of a new developed curve-face gear, and analyzed the compound transmission characteristics. [PERSON] [26] studied two ruled surfaces--the surface of a hyperbolic paraboloid and the tangent surface of a cylindrical helix, and summarized geometrical mathematical properties of both surface and their application. [PERSON] [27] proposed a unique gear set including a conical involute pinion and a mating face gear, which can transmit high torque at high speeds through an angle as required in a helicopter. [PERSON] [28] presented a new shaving method for double crowning, which used a variable pressure angle shaving cutter in a parallel gear shaving process. [PERSON] [29] studied the developed ruled surface of the face gear and provided an implicit equation, and the corresponding milling method was proposed based on the characteristics of the ruled surface. [PERSON] [30] put forward a method of tooth surface design and side milling of ruled line face gears. In general, in order to design high-performance gear transmissions or evaluate the meshing performance of gear pairs, tooth contact analysis (TCA) is required, which is a method of determining the tooth contact position, tooth contact area, and geometric transmission error through numerical simulation. However, in practical applications, gear pairsneed to withstand a certain load, and the load tooth contact analysis (LTCA) technology can better simulate the working performance under real working conditions. Through LTCA, the bearing contact imprint, tooth surface contact stress, tooth root bending stress, bearing transmission error, etc., of gears can be obtained, which is of great significance for shortening the cycle, reducing costs, and designing high-quality gears. [PERSON] [31] presented a new process of the TCA algorithm, which can reduce the calculation amount and speed up the calculation. [PERSON] [32] studied a new integration method for the generative design of face gears, which can improve the speed of CAD modeling and CAE finite element analysis. Based on the concept of the ruled surface of face gears, this paper proposes a method for machining ruled surfaces with conical cutters. So as to obtain the theoretical model of ruled surfaces, the mathematical model of the conical cutter will be defined, and the motion rules of the cutter will be derived according to the process of generating the ruled surface.