# Analytical and Graphical Profiling of Thread-Milling Cutters for Forming Internal Threads
## 8 Conclusions  
The paper proposes and evaluates a graphical profiling method referred to as the Tangent Circles Method. The method for graphical profiling of the tool surface of thread-milling cutters for internal threads is studied. It is based on the use of tangent circles and is implemented in a CAD environment. The method is validated by comparison with results obtained using the analytical method. Simulations with different parameters are also performed. The results show a very good match between the graphical and analytical approaches. The differences in the profile angle are minimal and within the tolerances for real manufacturing. The graphical method has several practical advantages. It enables quick profile visualization. It is easily performed using a standard CAD system. It requires fewer calculations. It is suitable for designers and technologists who work with tight deadlines and with limited resources. It can also be applied to non-standard tools where analytical approaches become too complex and difficult to apply and require high mathematical knowledge. The results demonstrate that the Tangent Circles Method is a reliable and effective technique for modeling complex tool geometries using CAD systems. Its ease of use and good accuracy make it a practical alternative to conventional analytical techniques. The method has strong potential for broader application in the design of advanced cutting tools and may be further implemented in automated tool-design systems or extended to support non-standard thread geometries. However, the method has its inherent limitations. It depends on the accuracy of the CAD software and the correct setting of the parameters. In case of errors in the geometry or insufficient resolution, deviations occur. The method does not take into account the influence of mechanical deformations, wear or sharpening errors. A positive aspect of the presented work is the analysis of the influence of the rake angle \(\gamma_{0}\) and the backing off \(\mathrm{K}_{0}\) on the profile of the tools. It is shown that these parameters have a significant influence on the profile of the cutting edges. By changing the rake angle, the height and inclination of the working surface change. With a combined change in \(\gamma_{0}\) and \(\mathrm{K}_{0}\), a visible deviation from the ideal profile is obtained. This shows that profiling should not be considered in isolation. It should be taken into account with the design and technological characteristics of the tool. The comparative analysis between the analytical and graphical methods confirms the reliability of the tangent circle approach. For different threads and tools, it has been found that the results of the two methods differ by less than one tenth of a degree in the profile angle. This is completely acceptable in design and production. The proposed graphical method of Tangent Circles is suitable to be used in manufacturing. It is accurate enough for real applications. It is easy to implement and understand. The authors support the idea that this method is also suitable for small and medium-sized companies where the necessary time cannot be allocated and there are no resources or capacity for full analytical modeling. However, if the task has increased accuracy requirements, or when profiling tools is characterized as critical, the proposed method should be used in combination with analytical analysis or with real verification of the results in production. Conceptualization, N.S. and A.I. ; methodology, A.I. and N.S. ; software, N.S. ; validation, N.S. and A.I. ; formal analysis, N.S. ; investigation, N.S. ; resources, N.S. ; data curation, N.S. and A.I. ; writing--original draft preparation, N.S. ; writing--review and editing, N.S. and A.I. ; visualization, N.S. ; supervision, N.S. ; project administration, N.S. ; funding acquisition, N.S. and A.I. All authors have read and agreed to the published version of the manuscript.