# A Unified Theory for 3D Gear and Thread Metrology
## 1 Introduction  
Gears and threads of all kinds are important elements in a broad variety of technical machinery. Their applications range from simple mechanical fasteners to key components found in all types of vehicles as well as in heavy-duty machinery and offshore energy systems. Whatever the scale, high-precision engineering is required to satisfy the narrow tolerances demanded in gear and thread production, tolerances that define the aims of manufacturing metrology. Current metrological challenges stemming from current trends in production technology can be summarized in five keywords: fast, accurate, reliable, flexible, and holistic [1]. Modern coordinate metrology systems are capable of gathering holistic information about the dimensions and surfaces of complex-shaped workpieces and can do so quickly, with high point density and with a good level of accuracy. However, standard evaluation procedures still reference single lines to represent the gear and thread geometry in the most relevant 2D sections. This is described for gears in the standards ISO 1328-1 [2] and AGMA 1012 [3], or in the corresponding VDI/VDE standards for profile and helix measurements [4] and pitch measurement [5]. The relevant definitions for threads can be found in ISO 5408 [6], ASME B1.1 [7], or DIN 2244 [8], while more specific measurement instructions for plugs and rings are described in VDI/VDA standards [9, 10]. This has been the state of the art for many decades, and it may be sufficient for quality control as long as the production method complies with the workpiece's kinematics and the manufacturing tolerances are not too tight. Beyond this, however, holistic evaluation procedures that assess the complete surface using one common model offer several advantages. Besides the obvious benefits of being able to determine deviations along the entire flank and of having sounder and more stable geometrical fitting parameters, holistic treatment allows us to find correlations between different measurands and to properly understand possible manufacturing errors. Moreover, modern production methods that donot follow the kinematic principles (such as five-axis milling) need holistic inspection for reliable quality assurance since line-based deviation analysis does not provide sufficient information on how to adjust machine tool parameter settings. For involute gears, an elementary approach to 3D descriptions was presented in [11]. In 1996, [PERSON] [12] was the first to introduce a holistic 3D model of involute gears to be treated as regular and freeform geometries in modern coordinate measuring technology. Detailed mathematical formulas can be found in [13]. Independently, [PERSON] proposed an alternative approach that also considered flank modifications [14; 15]. An extension of this work, in which the residuals of the fitted flanks are approximated by Chebyshev polynomials, can be found in [16]. As concerns screw threads, [PERSON] [17] developed an areal measurement strategy and holistic evaluation procedure based on an algorithm for non-linear least-squares approximation by Sourlier [18]. This technique has recently been verified by comparison to the established three-wire method [19]. Based on previous work that introduced a dedicated involute gear coordinate system [20], which was used to describe measurements in double flank contact [21] as well as profile and helix measurements [22], this article presents a universal model for helical machine elements. Defining a general class of helical machine elements allows metrological methods to be standardized for both gears and screw threads, as well as for any other geometry having an arbitrary transverse profile helically wound about the workpiece axis. This approach is advantageous as it permits general measurement strategies and evaluation algorithms to be easily adapted to special cases via a functional description of the transverse profile.