## 2 Novel Measurement Strategy
### GD&T (Geometric Dimensioning and Tolerance) Requirements  
The profile tolerances described in the GD&T multinational standard were used to define the boundaries for simplification. Figure 3 presents the input definition for measurement, the lines created at 15%, 50%, and 85% of the span and the trend line. Figure 4 shows an interpretation of profile tolerances within which the real shape should lie. No matter how many points are measured, all of them should lie within those two boundary lines. Each configuration should have its own calculations, but we can assume that a reasonable value should not exceed 2-3 mm for profile tolerance on cast surfaces. More details can be found under ASME [30] or ISO [31] standards. GD&T is a universal language both engineers and manufacturers use as it clearly communicates the main goal and functionality of the part definition. The part specifications should clearly and understandably describe the desired tolerance range from the ideal shape. In our research we used the standard profile of a line according to the ASME Standard.  
Figure 4: Ideal shape with knots with boundary of GD&T tolerance.  
Figure 3: Measuring sections with trend line.  
To be on the safe side, a trend line needs to be introduced, with an additional requirement that it maintains relation to all measured sections, no matter how many were defined. Figure 5a shows that the desired number of sections which help to meet our tolerance can always be found. This phenomenon occurs when all measurements are taken at the places where the curved line keeps within the boundaries of a tolerance zone. The overall boundary needs to be defined, using profile tolerance and aiming to ensure the situation in Figure 5b. Otherwise, our part can look like the exaggerated case shown in Figure 5c--the curvature can be beyond the desired tolerance and still be approved in accordance with the documentation prepared with no due diligence.  
This work proposes two criteria to locate a trend line, both of which are related to the part's mechanical construction. The first one is used when the part has a very thin trailing edge, especially for some large vanes, which tend to exhibit some flaws related to the complex curvature of the airfoil shape (Figure 6).  
Figure 5: (**a**) Potential error for sections measured separately; (**b**) trend line within the tolerance to assure all measured and not measured sections lie in the correct position; (**c**) potential defect in 3D.  
Figure 6: 2D section of two adjacent components with throat and thickness control measurements.  
The second criterion is related to the throat 3D line on a vane surface. The term 'throat' exists in the literature [32], but its real significance and full background may be found while studying some patents [33; 34; 35]. A throat width is the smallest distance between two points on two corresponding airfoil surfaces (Figure 6) presented on a 2D section. The throat area is the set of all throat lines between the inner and outer platforms. Point groups on each airfoil define a string which can be considered as a trend line for a vane.  
It is strongly recommended to use both these criteria during inspection. Firstly, trailing edge verification should be performed during casting measurements, and this was used in our work. Secondly, throat inspection on a concave (pressure) surface should be performed after coating (if this is one of the phases of manufacturing process) in order to control the real throat value there. The final throat should be inspected before assembly in the turbine, as this value significantly impacts engine efficiency.