## 3 Calculations  
To calculate a significant number of points for a CMM inspection, fundamental CAD math operations with spline interpretation need to be performed. Each section has been described as an extension of B-spline NURBS (non-uniform rational basis spline) in shape, shown in Figure 7. A B-spline can be understood as a collection of smaller Bezier three-degree curves beneath to achieve C2 continuity. The CMM base spline based on knot reduction was calculated following the method described in the work of [26; 27; 28]. Knot points are the most important and informative points and after simplification they indicate crucial shape changes. Figure 7: (**a**) Turbine guide vane control section defined by 104 Bezier curves before simplification. (**b**) Turbine blade control section defined by 127 Bezier curves before simplification. The original CAD section contains 104 smaller Bezier curves (Figure 7a), which is more than needed, as proved by other authors [15; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. It was rebuilt using knot removal methods and it was concluded that for CMM measurement the number of control points can be slightly decreased, reducing it to 20-30. This recalculation will produce a simplified B-spline with looser tolerance. Examples of point reduction and errors in shape are collated in Table 1. Different scenarios were tested, decreasing the number of points from 1000 into 5. It can be noticed that with around 25 points the spline is still of good quality, having only a few control points to inspect. The following spline simplification parameters were used: three-degree Bezier curves with C2 continuity and knot limitations with the maximum error less than 0.1 mm. Here, the widely known \"One to Ten Rule\" can be referred to. If the deviation and method accuracy in total is less than 10% of the desired tolerance, then the inspection method is safe and acceptable to use. The exercise was repeated with a different design for the airfoil shape--a turbine blade. The results are similar, and even for quite a different shape the observations regarding curve rebuild and point reduction are also similar (Figure 7b). The turbine blade section was used for contrast in the theoretical section of this article and will not be mentioned in further practical investigations. The average error chart based on Table 1 for both examples presented in Figure 8 proves that the distance error (distance between theoretical spline and the measured point) is on the same level for all points from 25 to 1000. To calculate the parameters, absolute values for error (inside/outside part material) and sample standard deviation were used. If we start from 100 points, 1000 will still be within the range, without any distortion--that is why standard deviation is 0. For 250 and 125 points, the rebuilt spline can be slightly different, with a small deviation. Using more than 50 points is not justified as this increases both costs and inspection time. Due to a rapid curvature change and being a very thin wall itself, the trailing edge can be removed from the enhancement and omitted during mass production. The horizontal trailing edge curvature is not so important from a mechanical perspective--it is slightly more valuable just to measure its thickness (Figure 6). A very critical part of the cast structure, which should be carefully verified to confirm overall surface smoothness, is a vertical curvature near the trailing edge--a trend line which lies on a throat area (Figure 6). It is critical for the part's performance and it has to be done simultaneously with the overall profile definition. Comparing with the original shape (the green dotted-dashed line) in Figure 9 and calculating deviations based on 25 points (the blue straight line), any deviation impact can be omitted as the spline average error will not affect the part shape at all (Table 1). Understanding the error resulting from an adjusted spline defined by 25 control points should be considered during engineering.