# Options on Tooth Profile Modification by Hob Adjustment
## 1 Introduction  
Hobbing mills represent a special category of high-performance tools in the field of engineering, with the main focus being on the production of gears and other complex machine parts, such as splined shafts, sprockets and other components with special screw surfaces. Their effectiveness lies in the combination of a number of cutting edges and their continuous, smooth engagement, which significantly minimizes wasted time and maximizes the productivity of the process. These tools are characterized by a high degree of variability in their use, which means that with the same tool it is possible to produce gear segments of different shapes and sizes, with a variable number of teeth and a variable pitch, with or without gear correction, and also in relation to worm wheels. The method of producing teeth using a shank or disc modular milling cutter is more expensive compared to the hobbing mill method, because when using wheels with the same module and a different number of teeth, the profile of the tooth gap changes, which also results in a more complex profile for the modular milling cutter, which complicates the production and grinding. Gears are essential components for the transmission and transformation of mechanical motion and energy in machines and mechanisms. Their curved shapes require specific mathematical tools for the design of production tools. Although they have been around for centuries, they are constantly being improved. Publications dealing with the development of efficient gear manufacturing tools remain relevant. This text deals with the investigation of the basic shape of the tool in terms of kinematic schemes for material separation. Analytical methods for determining the tool shape require knowledge of analytical geometry in space and the differential geometry of curves and surfaces, which are investigated using differential calculus. Within the framework of this topic, several studies have been published. [PERSON] et al. [1] aimed their research at the experimental verification of the calculation algorithms, taking into account the wear of the milling cutter teeth in individual generating positions. The replacement of the classical analytical solution for the machining of involute gears with a general mathematical model was dealt with by [PERSON] et al. [2]. In their work, they dealt with the creation of a general mathematical model by generating two parameters using a discrete method to increase the robustness of the method. They designed a numerical algorithm by transforming multiple cutting edges with respect to the workpiece of the gear wheel based on generated motions. Subsequently, they introduced the derivation of instant contact points using the implicit time sequences of the profile points. In the article by [PERSON] et al. [3], the possibilities of gear design using the method of discretization of the gear tooth flanks were analyzed, while instead of an analytical approach, they proposed dividing the flank of the tooth into an infinite number of local involute segments. As an advantage of this method, they presented its relative simplicity compared to the analytical solution. The method was implemented and verified when applied to both globoid and involute gears. [PERSON] et al. [4] contributed to the previously mentioned studies on this topic by focusing on creating an algorithm suitable for CAE simulations, as in addition to the produced gearing, they also monitored the shape and volume of the resulting chip. Within the outputs, the authors present the developed algorithm that was embedded into a CAD environment. They proved that the algorithm is suitable for predicting cutting forces, tool stress and wear, as well as optimizing the parameters of the production process. The production of gears using hobbing mills enables the production of gear segments with a consistent profile, which reduces the difficulty in setting up and contributes to the efficiency of the production process. Gears are an integral part of many mechanical systems and devices.