## 1 Introduction  
Faced with increasing demands for precision and reliability in both the mechanical and energy industries, engineers and scientists strive to develop measuring and production methods that enable the achievement of these standards. Mechanical and energy products, such as turbines, pumps, and compressors, require strict adherence to geometric specifications to ensure their energy efficiency and reliable operation [1]. In the functional requirements of modern energy products, increasing complexity is noticeable, which implies the necessity of adhering to strict criteria regarding the deviation of the actual geometry from the nominal [2]. The efficiency of these products depends on the precise matching of surfaces, where the partition surfaces enable the controlled flow of the working medium through the machine, and non-contact seals, such as sliding rings, minimize energy losses. Therefore, the selection of an appropriate production process to maintain the high quality of the final product is also essential [3; 4; 5]. The dividing and sealing surfaces in flow energy machines such as turbines, pumps, and compressors play key roles in maintaining energy efficiency as well as ensuring the reliability and safety of these devices' operation. This means that decisions regarding the production process, from the choice of tools and materials to measurement sampling strategies, are crucial for achieving optimal results [6]. Partition surfaces divide the machine into different sections or stages, allowing for controlled flow of the working medium (e.g., steam, gas) through the machine. They enable the optimal use of the energy contained in the working medium at each stage of the process. In turbines and other flow machines, the direction of the flow of the working medium is crucial for efficiency [7; 8]. Dividing surfaces are used to direct this flow in a way that maximizes energy efficiency by reducing losses caused by turbulence; for instance, by maintaining tightness between different parts of the machine, dividing and sealing surfaces also help in maintaining the appropriate working pressure in each section [9; 10]. To ensure the continuity of the flow of a medium (liquids or gases) while maintaining physical separation between the moving parts of a machine, slip rings are used. This solution is for non-contact seals, which find application in many areas of industry. Slip rings for non-contact seals are used in various types of pumps (such as vacuum pumps and water pumps) and compressors, where they provide the necessary sealing with minimal impact on performance. They are also used in the chemical industry in reactors and pumps for aggressive chemical substances, as well as in mixing systems. Thanks to the use of materials with high resistance to corrosion and chemicals, slip rings withstand difficult operating conditions, ensuring the continuity of production processes. Due to their unique construction and the materials that they are made of, slip rings minimize friction and wear, which translates into long life and reliability of the sealing system. Precise dimensions of the slip rings are essential to ensure optimal fitting and sealing [8; 9; 11]. Measurements of shape errors are a crucial element of machine part inspection, complementing linear and angular dimensions with a geometric specification of the product. Traditional measurement methods, using universal measuring equipment such as dial indicators or interferential plates, have a limited application. Among modern methods, we can distinguish between measurements using devices with a rotary measuring table, the use of profilometers, interferometric studies, or examinations using 3D machines [1; 12; 13; 14]. Among all industrial inspection methods, tactile measurement using coordinate measuring machines or tool control numerical machines is the fundamental technique due to its high accuracy, reliability, and versatility [15].