## 1 Introduction  
Computerized simulation of gear tooth contact is a significant part of gear design. It allows to assess all the processes of gearing at the design stage, which may include misalignment and transmission errors (TE), tooth damage, noise, and vibration. This field is well-studied by [PERSON] [1], who developed seminal methods for the so-called tooth contact analysis (TCA) and local synthesis, calculating contact between two teeth for machine tool settings. However, the key drawback of the model is the need for estimation of the position of the surface contact in order to achieve an accurate solution, which requires substantial computational power; otherwise, the numerical computations process suffers from convergence issues. Various researchers have proposed other computational methods of tooth contact analysis in order to resolve the stated and other problems associated with the method. [PERSON] [2] suggested a fast-geometric approach to calculate the contact pattern without the need to take into account the curvature of the surface. The study showed that the instantaneous contact area is estimated using the surface intersection procedure that mimics the removal of the marking compound during meshing. In addition, [PERSON] and [PERSON] [3] proposed that the use of normal vectors to evaluate the positions of the contact points according to the surface location can be avoided solely by using a numerical solution vector. [PERSON] [4] suggested a discrete optimization approach to define the minimum distance points between two surfaces to solve the contact problem. Later, a new type of fundamental two-dimensional tooth surface contact equations was suggested for the two-dimensional problem (i.e., spur parallel axis gearing) by [PERSON] et al. [5, 6], which reduced [PERSON]'s model with five typically nonlinear equations with five unknowns and one independent parameter to a system with only two equations with two unknowns and one independent parameter. Often, theoretical conjugate models of gears do not correspond to the practical application due to deliberate modifications of the tooth or to manufacturing errors [7]. Underthese conditions, the use of the classical theory of gears can lead to erroneous results that do not agree with reality. The mesh analysis of these non-conjugate gears is necessary in order to account for the resulting non-uniformity in motion transmission in addition to the changed direction of contact. Therefore, over time, various models for non-conjugate action gears have been proposed. One of them was created by [PERSON] and [PERSON] [8], who built a model for helical gears tooth meshing with deformed surfaces designed to minimize TE and account for production errors and complex load distribution. By applying this approach, the authors were able to enhance and adjust the quality of the meshing by accounting for load distribution. During the power transmission of spur and helical gears, various errors such as machining errors, misalignment of the assembly, and other factors may cause noise and vibration during the operation. For a long time, gear alignment problems (or misalignments) have concerned engineers. Gear-pair misalignments can affect both the geometric and physical performances of the tooth flanks. Determining the loaded tooth contact performance with misalignments is crucial for improving strength and stabilizing the gear transmission system. In order to accomplish an assessment of the contact characteristics without edge contact and concentrated stresses, low vibration, and acceptable transmission error, research on loaded tooth contact analysis (LTCA) has been conducted [9; 10; 11]. The load distribution calculations proposed by [PERSON] [9] showed that LTCA provides a more realistic contact pattern and contact pressure than the elliptical one. [PERSON] et al. [10; 11], by using LTCA, were able to obtain more accurate results of the tooth contact pattern by determining whole load distribution over the entire tooth surface.