Evaluating the geometric characteristics of cylindrical features

Precision Engineering 27 (2003) 195–204

Evaluating the geometric characteristics of cylindrical features S. Hossein Cheraghi a,∗ , Guohua Jiang b , Jamal Sheikh Ahmad a a

Department of Industrial and Manufacturing Engineering, Wichita State University, Wichita, KS 67260-0035, USA b Cessna Aircraft Company, Wichita, KS 67277, USA Received 18 September 2001; received in revised form 10 September 2002; accepted 31 October 2002

Abstract This paper presents mathematical models and efficient methodologies for the evaluation of geometric characteristics that define form and function of cylindrical features; namely cylindricity and straightness of median line. These two problems have similar structures and can be solved by comparable procedures. Based on the proposed methodologies, the cylindricity error evaluation can be performed using any of the following criteria: the least squares cylinders, minimum circumscribed cylinders, maximum inscribed cylinders or minimum zone cylinders. The procedures have been tested for accuracy and efficiency. The results indicate that they provide accurate results quickly. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Cylindricity error; Straightness error; Coordinate measuring machines; Median line

1. Introduction A great majority of mechanical parts comprise of cylindrical features (internal and external). Two basic geometric characteristics that are used to control form and function of cylindrical features are cylindricity and straightness of an axis. As defined by the ASME standard on geometric dimensioning and tolerancing (GD&T), cylindricity refers to a condition of a surface of revolution in which all points of the surface are equidistant from a common axis. Cylindricity error is the radial distance between two coaxial cylinders (one circumscribed and one inscribed) containing the cylindrical profile. Similarly, straightness is a condition where an element of a surface, or an axis, is a straight line. Straightness error of an axis is the radius of the smallest size cylinder within which the considered derived median line must lie [1]. Significant error associated with these characteristics may result in the failure or inadequate functioning of the corresponding part. Accurate measurement of these errors is not a trivial task due to the three dimensional nature of the characteristics. The methods for the evaluation of form errors can be divided into the intrinsic datum method and extrinsic datum method. In the intrinsic datum method, points on the surface of the part are used as a datum. The most commonly used intrinsic techniques for cylindricity evaluation are diametrical measurement, V-block measurement and bench center mea∗ Corresponding

author. Tel.: +1-316-978-5915; fax: +1-316-978-3742. E-mail address: [email protected] (S.H. Cheraghi).

surement [2]. The intrinsic methods are generally inaccurate due to the existence of multiple sources of error associated with the measurement process. In the extrinsic datum method, an external member is used as the datum reference. An example of an extrinsic method is when data on the surface of a cylindrical feature is collected and used in a computer program to calculate the geometric error. Extrinsic methods are more time consuming to apply but they are more accurate. With the existence of Coordinate Measuring Machines (CMMs) that can provide accurate coordinate information; extrinsic methods can accurately evaluate the error if a good evaluation algorithm is available. Several criteria have been proposed for the evaluation of cylindricity error. They are least squares cylinder (LSC), minimum circumscribed cylinder (MCC), maximum inscribed cylinder (MIC) and minimum radial separation (MRS). The two common extrinsic methods for the evaluation of straightness error of a median line are the least squares (LS) and the MCC methods. These techniques are discussed in more details in the next section. Most CMMs utilize the LS technique due to the simplicity of its application. This technique, however, is an approximation technique and overestimates the error resulting in the rejection of good parts. The mathematical evaluation techniques that are based on the MRS criterion are either complex or lack an algorithmic approach to find the optimal solution. Furthermore, the lack of efficient algorithms based on MIC and MCC has made their usage uncommon. In this paper, we present mathematical modeling, and simple but effective algorithm for the evaluation of cylindricity

0141-6359/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 6 3 5 9 ( 0 2 ) 0 0 2 2 1 - 0

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error. Based on the proposed algorithm, the cylindricity error can be assessed using any of the above-mentioned evaluation criteria (namely LSC, MCC, MIC or MRS). We then extend the procedure for cylindricity to evaluate the straightness error of an axis. These two problems have similar structures and can be solved by procedures that are similar in nature. The proposed procedure can be used to evaluate the straightness error of an axis using the MCC or LS technique. Experimental results indicate that the algorithms are fast and the results they provide are very accurate. The rest of the paper is divided into two major parts. In the first part the mathematical modeling, solution algorithm, and performance results for the cylindricity error evaluation is presented. The straightness error evaluation problem is presented in the second part.

2. Review of background material and the related literature As mentioned in the previous section, the most common extrinsic methods for the evaluation of cylindricity error are MRS cylinders, LSC, MIC, and MCC. All these techniques try to find two coaxial cylinders at minimum separation within which the surface of the measured feature is to fall. The radial difference between these two cylinders is used as a measure of cylindricity error. The difference between these techniques is in the method of finding the location and orientation of a common axis. With the LSC method, a cylinder is fitted to the profile using the LS method. The axis of that cylinder is used as the common axis for the smallest circumscribed and the largest inscribed cylinders. With the MIC method, the axis of the largest cylinder that can be contained inside the profile without including any part of the profile is found. That axis is then used to find the smallest cylinder that contains the profile. With the MCC method, the smallest cylinder that contains the profile is first found. The axis of that cylinder is then used to find the largest inscribed cylinder. Among these methods, only MRS is based on the ASME Y14.5-1994 standard on GD&T. The most common extrinsic methods for the evaluation of straightness error of a median line are the LS method and the MCC method. Given a set of data points representing the median line, with the LS method a spatial line is fitted to the data points using the LS method. The fitted line is then used as the axis of the smallest circumscribed cylinder whose diameter would define the straightness error. With the MCC method, the smallest circumscribed cylinder that contains the data points is found and its diameter is defined as the straightness error. The MCC method is based on the ASME Y14.5 standard on GD&T. Most of the articles found in the literature, related to the topic of this paper, deal with the evaluation of cylindricity error. Very few articles can be found which specifically deal with the evaluation of straightness error of medial line.

Wang [3] proposed a general-purpose algorithm for constrained nonlinear optimization problems for minimum zone evaluation of form tolerances. The idea is to use techniques like the sequential quadratic programming method with the hope that optimal solution to the quadratic programming problem can be obtained in a finite number of iterations. Shunmugam [4] proposed a method based on the minimum average deviation for evaluating form error of lines, circles, planes, cylinders and spheres. He used linearization and the simplex search method to minimize the sum of the absolute deviation values. This method is not based on MRS criterion. Carr and Ferreira [5] presented nonlinear optimization models for calculating cylindricity and straightness error of a median line which they solved using successive linear programs. This is similar to the method presented by Wang [3]. The linearization process could result in an approximate solution rather than an optimal solution. The efficiency and robustness of the procedure depends on how far the initial values are from the final optimal solutions. A strategy based on geometric representation for minimum zone evaluation of circles and cylinders was proposed by Lai and Chen [6]. The strategy employs a non-linear transformation to convert a circle into a line and a cylinder into a plane, and then uses a straightness or a flatness evaluation schema to obtain minimum zone deviation for the feature concerned. This is an approximation strategy to the minimum zone circles or cylinders. Roy and Xu [7] proposed a computational geometry-based technique to generate a pair of concentric cylinders for checking the cylindricity tolerance and a center axis for the verification of orientation tolerances of a cylindrical feature. In this method, the cylindrical surface is divided into several cross-sections normal to the CMM’s local z-axis. Then 2-D convex hull and voronoi diagrams are used to generate pairs of concentric circles, and these circles are used to find a pair of concentric cylinders to create the minimum zone required for form tolerance analysis. This method assumes that the orientation of the cylindrical features is around the z-axis of measurement. Lai et al. [8] used genetic algorithms to evaluate cylindricity error. The numeric oriented genetic operator is employed as a basic representation for error modeling. The implementation of this method may be complex.

3. Cylindricity error evaluation As defined by the ASME Y14.5M-1994 standard on GD&T, “a cylindricity tolerance specifies a tolerance zone bounded by two concentric cylinders within which the surface must lie.” Fig. 1 shows an example of a cylindricity tolerance application and its tolerance zone definition. [1] The cylindricity specification in the figure indicates that the surface of the cylinder is to fall within two coaxial cylinders having a radial difference of 0.20.

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where e is the circularity error calculated according to a selected evaluation criterion (i.e. MRS, LSC, MIC, or MCC). The model in F1 searches for rotation angles (θ x , θ y ) through which if the measured points are rotated and then projected onto the x–y plane, the circularity error of the projected points is minimized. Mathematical definition of circularity error function, e(Piz , x, y), varies based on the evaluation criterion. If the minimum zone criterion is used, e(Piz , x, y) would be defined as follows: e(Piz , x, y) = max d(Piz , x, y) − min d(Piz , x, y), i

Fig. 1. A cylindricity application and its tolerance zone definition.

3.1. Mathematical modeling The proposed methodology for the evaluation of cylindricity error uses a circularity error evaluation procedure as a subroutine. The developed methodology is constructed based on the following argument. If we rotate a cylinder, which has a perfect form, such that its axis parallels the z-axis and then project all points on the surface of the cylinder onto the x–y plane, the projection points will form a perfect circle on the x–y plane. It is clear that this circle has a circularity error equal to zero. As the orientation of the axis of the cylinder is changed, the projection points of the surface would no longer form a perfect circle, thus, having a circularity error greater than zero. Based on this logic, optimal cylindricity error can be defined as the minimum value of circularity error of the project points as the cylindrical feature is rotated around the x and y axes. Based on this discussion, let n

number of points sampled from the surface of the cylindrical feature; Pi measured point i, i = 1, . . . , n; θx, θy rotation angles about x and y axes, respectively (decision variables); Pir = R · Pi measurement point Pi after rotation where   cos θy 0 sin θy R =  sin θx sin θy cos θx −sin θx cos θy  −cos θx sin θy sin θx cos θx cos θy is the rotation matrix around x and y axes;

Piz

projection of Pir onto the x–y plane; coordinates of the center of the two concentric circles containing the projected points Piz (decision variables).

x, y

The mathematical model of the cylindricity error evaluation problem can be stated as follows. Optimal value of ρ in F1 determines the cylindricity error. (F1) min ρ(θx , θy )

(1)

θx ,θy

where ρ(θx , θy ) = min e(Piz , x, y), x,y

i = 1, . . . , n.

(2)

i

i = 1, . . . , n

(3)

where d(Piz , x, y) is the distance from Piz to (x, y). If the LS method is used, e(Piz , x, y) would be defined as follows: n  (4) [d(Piz , x, y) − r(x, y)]2 e(Piz , x, y) = i

where r(x, y) is the radius of the LS circle fitted to points Piz , and d(Piz , x, y) is as defined before. Similar mathematical representations can be written for other evaluation criteria such as MIC and MCC. 3.2. Solution methodology The solution methodology consists of two loops: an outer loop and an inner loop. The outer loop uses an efficient search algorithm to find values of rotation angles such that the objective function in (1) is minimized. The rotation angles are then input to the inner loop. The inner loop calculates the value of circularity error for the given rotation angles. The interaction between the inner and the outer loop continues until no further improvement in the value of circularity error could be observed. In the following subsections, the procedures for each loop are discussed in details. 3.2.1. Outer loop The task in the outer loop is to search for optimal values of θ x and θ y such that the objective function in formulation F1 is minimized. For fixed values of θ x and θ y , ρ(θ x , θ y ) is calculated. The values of θ x , θ y are then changed in the direction that improves the objective function value in (1). The search continues until no improvement in the objective function value can be found. The procedure is as follows. 3.2.2. Algorithm outer loop Step 1. Given a step size ϕ, a very small number ε, and initial rotation angles θ x and θ y . Step 2. Call the inner loop procedure to calculate circularity error (eij ) at θx + ϕ ∗ (i − 1), θy + ϕ ∗ (j − 1), where i, j = 0, 1, 2, respectively. Step 3. If e11 = mini.j eij , go to Step 4. Otherwise, set θx = θx + ϕ ∗ (i  − 1), θy = θy + ϕ ∗ (j  − 1) where i and j are

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the index values for which the objective function is minimal (ei  j  = mini,j eij ), and go to Step 2. Step 4. If ϕ ≤ ε, then stop; θ x , θ y and ρ(=eij ) give the desired solution; otherwise set ϕ = (ϕ/2) and go to Step 2.

the MRS and LSC techniques have been implemented. Many procedures exist for the calculation of the MRS circles. Wang and Cheraghi [9] proposed an efficient algorithm for calculating minimum zone circles. Their algorithm has been utilized in the inner loop. Similarly, there are existing methods for the calculation of circularity error based on the LS criterion (Murthy [10], Kim and Kim [11]). We have developed an iterative algorithm to calculate circularity error based on the least-squares technique (the detail of this algorithm is not presented in this paper). The procedure is very effective and hence is used in the inner loop of the proposed algorithm. There are also efficient procedures for the evaluation of circularity error based on MCC and MIC. Elzinga and Hearn [12] and Preparata and Shamos [13] proposed efficient algorithms for the calculation of MCC and MIC, respectively. These procedures can be easily implemented. Based on the above discussions, the procedure for the evaluation of cylindricity error is summarized by the flowcharts shown in Fig. 2.

The input to the outer loop algorithm is the initial values of rotation angles (θ x , θ y ). Two techniques were used to derive the initial rotation angles: the LS technique and the partition method. While both procedures provided good initial solutions, LS was preferred because of its simplicity and speed. With this technique, the direction of the best-fit axis is used as an initial solution (i.e. initial rotation angles). The LS technique is widely used in industry for the evaluation of form errors because of its ability to provide good solutions quickly. It is also used by researchers as the method of choice to generate initial solutions for their optimization procedures to evaluate form errors. This is because the solution provided by the LS technique is in the neighborhood of the optimal solution and hence the most suitable and logical initial solution.

3.3. Implementation and performance evaluation

3.2.3. Inner loop The inner loop finds the circularity error of the projected points of the surface of the cylinder based on the selected criterion (i.e. MCC, MIC, LSC and MRS). Any of these techniques can be used in the inner loop. In this research, only

The proposed procedure for the evaluation of cylindricity error was implemented using the C++ programming language. Two measures of performance were used to evaluate its performance: the ability to provide accurate solution and the computational efficiency. The datasets used in this

Fig. 2. Cylindricity error evaluation procedure.

S.H. Cheraghi et al. / Precision Engineering 27 (2003) 195–204 Table 1 Comparison of the cylindricity error results with the published results Cylindricity error

Dataset 1 [5] Dataset 2 [5] Dataset 3 [5] Dataset 4 [8]

Proposed procedure

Published results

0.00100 0.18395 0.00942 0.002788

0.00100 0.18396 0.00941 0.002788

199

The relative evaluation of the computational efficiency against other existing algorithms was not possible due to the fact that either no data on the speed of their procedures have been provided by other authors or that the procedures were run on different platforms and hence not comparable on the same basis.

4. Straightness of median line evaluation were taken from published articles and from the measurement of real parts (see Appendix B for datasets from published articles). To test the procedure’s ability in generating accurate solutions, it was applied to a set of data taken from the literature, using the MRS criterion in the inner loop [5,8]. The results were then compared with published results as shown in Table 1. As the numbers in the table indicate, the developed procedure has been able to accurately calculate the cylindricity error values. In order to test the computational efficiency of the proposed procedure, a computer program was developed to randomly generate data under different conditions. The variables that were considered in generating the random numbers were the radius (R) and height (H) of a cylindrical feature, and number of points (n) representing problem size. A large number of datasets were generated using R = 10, H = 100 with n ranging from 25 to 250 points with 25 points increment. The procedure running on a PC with a 450 MHz Pentium processor was tested on the generated datasets. Graphical representation of the results shown in Fig. 3 indicates a linear relation between problem size and computation time. For a dataset containing 50 points it took the procedure, on average, less than 0.5 s to arrive at a solution. These results provide a strong evidence to prove the computational efficiency of the procedure in absolute terms.

When straightness is applied to a cylindrical feature of size, it controls the derived median line of the feature’s actual size that is a spatial line. The derived median line is generated by connecting the centers of circular cross sections along the length of the part. In this case, the tolerance zone is a cylinder within which the median line should fall. Fig. 4 shows an example of a straightness specification applied to a feature of size. The specification requires that the derived median line of the feature must lie within a cylindrical tolerance zone of 0.04 diameter, regardless of feature size. The straightness error is the diameter of the smallest cylinder containing the generated median line [1]. 4.1. Mathematical modeling The straightness error of a median line can be evaluated using a methodology similar to that of cylindricity.

Fig. 4. Application of straightness to a feature of size [9].

Fig. 3. Computational efficiency of the proposed algorithm.

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As described previously, in order to evaluate straightness error of a median line, a cylinder with minimum diameter that contains all the points representing the median line should be found. The problem of evaluating straightness error of a median line can be transformed to a 2-D minimum circumscribed circle problem. If we rotate a perfect spatial line such that it parallels the z-axis and then project all points on this spatial line onto the x–y plane, the projection points will form a point (the minimum circumscribed circle containing all the projection points has a diameter of zero). Hence, the straightness error of a spatial line can be obtained by calculating the diameter of the minimum circumscribed circle containing the projection, onto the x–y plane, of points representing the median line. Based on this idea, the following non-linear programming model has been developed for the evaluation of the straightness error of a spatial line: (F2) min ρ(θx , θy )

(5)

θx ,θy

The optimal objective function value is the diameter of the MCC that contains the median line (straightness error). 4.2. Solution methodology The solution methodology for this problem is similar to that of the cylindricity error evaluation as is shown in the flow chart in Fig. 5. It is a two-loop procedure. The outer loop is the same as that of cylindricity error evaluation procedure. It finds rotation angles θ x , θ y so that the axis of the cylinder containing the derived median line is parallel to the z-axis. The inner loop finds the minimum circumscribed circle that contains all the points (representing median line) projected on the x–y plane. Any method for finding minimum circumscribed circle can be used in the inner loop. In our procedure, we have used Elzinga and Hearn’s algorithm [12]. As with the cylindricity problem, the LS method may be used to find initial rotation angles. 4.3. Performance evaluation

where ρ(θx , θy ) = min e(Piz ), x,y

i = 1, . . . , n

(6)

and “e” is diameter of the minimum circumscribed circle of the projected points onto the x–y plane.

The proposed procedure for the evaluation of straightness error of a median line was implemented into a computer program and was tested for validity and efficiency. To test the validity of the proposed procedure, it was applied to datasets

Fig. 5. The proposed procedure for the evaluation of straightness error of a median line.

S.H. Cheraghi et al. / Precision Engineering 27 (2003) 195–204 Table 2 Comparative analysis of results for straightness of a spatial line Straightness error

Dataset 5 [5] Dataset 6 [5]

Proposed method

Published results

0.015089 0.899967

0.0151364 0.9001114

taken from a published paper [5]. All the datasets are listed in Appendix B. The comparative results are shown in Table 2. As shown in the table, the results provided by the procedure are the same or slightly better than the published results. To test the computational efficiency of the proposed procedure, a computer program was developed to randomly generate data under different conditions. A large number of datasets were generated using a data-generation program similar to

201

that discussed in the cylindricity error evaluation section. The developed program was run on a PC with a 450 MHz Pentium processor. The graphical representation of the results is shown in Fig. 6. As shown in the figure, the computation time increases with an increase in the number of points but with a very small slope. The computation time is insignificant (<0.2 s) even with 250 points in the dataset.

5. Convexity discussion The procedures discussed in the previous sections would generate optimal solutions only if the functions shown in F1 and F2 are convex. These functions involve non-convex sinusoidal forms and are not convex. A graphical representation of the function in F1 is shown in Fig. 7.

Fig. 6. Computational efficiency of the spatial line procedure.

Fig. 7. Graphical representation of ρ(θ x , θ y ).

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In practical industrial applications, the produced cylindrical features are expected to have small variations from their nominal values. As stated in the previous sections, the LS method is used to find initial rotation angles. The initial rotation angles provided by the LS method is in the neighborhood of the optimal solution. For this reason, the behavior of functions in F1 and F2 for small rotation angles becomes important. To study this behavior, an empirical study was carried out and the results are shown in Appendix A. As shown in the Appendix, the results indicate that it is safe to conclude that both functions are convex within a practical range of rotation angles, and hence the developed procedures should provide an optimal or a near optimal solution. 6. Conclusions This paper presented efficient procedures for the evaluation of cylindricity error, and straightness errors of medial line. These two problems have similar structures and can be solved by similar procedures. The results from the evaluation of the procedures indicate that they are capable of providing accurate results quickly. An important advantage of the proposed cylindricity procedure is that it allows different methods for the evaluation of cylindricity error such as MCC, MIC, MRS or LSCs be used.

Fig. 8. Graphical representation of ρ(θ x , θ y ) in F1 for two different ranges of rotation angles.

Appendix A. Convexity discussion As it was mentioned in Section 5, for the procedures to provide optimal solutions it is important that functions in F1 and F2 are convex within a practical range of rotation angles (θ x and θ y ). This appendix presents graphical analysis of these functions. The analysis was carried out for six different ranges of rotation angles: ±1, ±5, ±10, ±30 and ±45◦ . Even though the results presented in this section hold true for all ranges of rotation angles that were tested, we only present the graphical representations for two of these ranges (±1 and ±5◦ ). Fig. 8 shows the three-dimensional graphs of ρ(θ x , θ y ) in F1 for these ranges when the MRS circles method is used in the inner loop. Similarly, Fig. 9 shows the three-dimensional graph of ρ(θ x , θ y ) in F2 for the same ranges of rotation angles. These graphs indicate that the functions behave in a convex manner within the specified ranges. It is only when the angles are increased beyond the 90◦ range that the models exhibit non-convex behavior. Based on this analysis, it would be safe to conclude that the functions are convex within a practical range of rotation angles, and hence the developed procedures should provide an optimal or a near optimal solution.

Fig. 9. Graphical representation of ρ(θ x , θ y ) in F2 for two different ranges of rotation angles.

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Appendix B. Datasets for testing the procedures Dataset 1 40 points 30 1.05636 −29.366428 28.698913 −12.893256 −22.315616 14.611799 28.323328 −14.26238 −22.304724 −26.057635 −25.432673 17.043966 25.129457 −25.917641 25.362190 −2.34494 −2.62016 −20.170149 29.952444 30.001 1.056395 −29.367407 28.699869 −12.893685 −22.316359 14.612286 28.324273 −14.262855 −22.305468 −26.058504 −25.433521 17.044534 25.130294 −25.918505 25.363036 −2.345018 −2.620248 −20.171721 29.953442

0.001475 −29.981396 −6.132937 8.739131 −27.088078 20.050269 −26.201056 9.888835 −26.39288 20.062385 14.866056 −15.911603 −24.688119 16.386287 15.108801 −16.03271 29.908214 −29.88536 −22.206503 −1.68852 0.001475 −29.982395 −6.133141 8.739422 −27.088981 20.050938 −26.201929 9.889165 −26.393767 20.063053 14.866552 −15.912134 −24.688942 16.386834 15.109304 −16.024245 29.909211 −29.886356 −22.207243 −1.688577

7.892267 27.519008 13.137551 40.731883 56.081574 31.164982 2.074327 31.782012 0.461891 4.010534 41.206363 55.826190 31.615727 39.235138 42.071436 45.731882 2.847871 19.694054 45.384629 21.92032 7.892267 27.519008 13.137551 40.731883 56.081574 31.164982 2.074327 31.782012 0.461891 4.010534 41.206363 55.82619 31.615727 39.235138 42.071436 45.731882 2.847871 19.694054 45.384629 21.92032

Dataset 2 20 points 60.051121 −57.932024 57.432130 55.022756 29.180100 −58.861558 −44.597179 −23.247383 34.041568

0.002953 15.399312 17.488707 −23.936632 −52.423113 −11.113569 40.113733 −55.406652 −49.309081

3.946134 15.983017 20.365942 11.505062 1.037163 20.134482 2.005267 17.669299 15.807863

203

Appendix B (Continued ) −34.084135 50.684216 57.318676 −40.408130 −39.838370 −10.261352 53.919844 −8.540012 −59.369089 −38.029817 47.946099

−49.427745 −32.022045 17.619539 −44.485701 44.994386 −59.146784 26.493193 59.442972 8.361285 46.404843 −35.925380

12.479981 22.865941 22.082457 22.692315 7.411167 22.600675 18.949042 13.092342 7.133233 4.995216 27.276243

Dataset 3 40 points −11.820859 42.403448 10.366902 18.527457 23.930322 66.363729 −3.608026 75.507564 48.919097 65.713317 46.632786 13.598993 84.570573 2.322453 82.820384 3.553158 −5.898713 30.009532 −3.793621 58.357492 33.207329 34.46129 −26.871029 −4.153639 22.371000 67.398986 79.257377 −37.543275 49.576671 96.781947 −18.623157 58.416292 48.408528 31.694971 −18.366214 81.087477 57.311572 68.59397 89.036231 3.141412

50.421254 −6.693162 80.249947 61.577469 23.878386 0.636729 −24.493246 20.208045 55.614254 2.841028 80.517454 83.519129 18.219363 −10.802862 38.516367 75.111087 21.39033 −24.696147 −14.263808 87.161327 64.844079 41.806234 3.103967 67.427229 47.845956 16.520701 49.418921 31.718373 65.965076 53.421231 23.988046 −4.557784 15.833662 −2.169579 2.837799 11.573666 −9.09605 33.580936 21.72231 52.730721

−15.817382 56.567707 26.965969 −13.680418 −41.820643 49.246025 39.678687 6.298139 −13.266609 3.498858 4.866333 30.375 28.224203 51.268799 9.148307 30.738097 60.097056 35.870356 46.897322 11.960644 −10.665479 94.623903 39.48246 23.451422 88.060867 79.062822 4.727043 8.573268 −6.501629 22.908004 47.691608 48.525368 81.511728 63.538387 46.415679 46.319607 38.123767 −6.118165 35.086999 67.919265

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Appendix B (Continued )

Appendix B (Continued ) Dataset 4 24 points 11.094300 5.094000 −6.906300 −12.906500 −6.906300 5.094000 10.954600 4.954400 −7.045900 −13.046100 −7.045900 4.954470 10.815000 4.814800 −7.185500 −13.185800 −7.185500 4.814900 10.675400 4.675200 −7.325300 −13.325400 −7.325200 4.675200

0.452200 10.845000 10.843900 0.449800 −9.942900 −9.941800 0.522000 10.914800 10.913700 0.519600 −9.873100 −9.872000 0.591800 10.984600 10.983500 0.589400 −9.803300 −9.802200 0.661600 11.054400 11.053300 0.659200 −9.733500 −9.732300

65.232800 65.076500 65.008900 65.089700 65.054000 65.221600 75.231600 75.075200 75.077000 74.896400 75.052800 75.220400 85.230400 85.074000 85.064100 84.895200 85.051600 85.217100 95.229100 95.072800 94.907700 95.094000 95.050400 95.217900

0 100 200 300 400 500 600 700 800 900

−0.003 0.009 0.015 0.025 0.029 0.036 0.042 0.051 0.059 0.072

0.005 −0.012 −0.025 −0.038 −0.040 −0.048 −0.059 −0.062 −0.074 −0.088

Dataset 6 22 points 0 254 508 762 1016 1270

0.410 0 −0.108 −0.170 −0.112 −0.068

0 0.124 0.205 0.306 0.352 0.387

Dataset 5 10 points

1524 1778 2032 2286 2540 2794 3048 3302 3556 3810 4064 4318 4572 4826 5080 5334

−0.050 −0.150 −0.302 −0.286 −0.220 −0.180 −0.148 −0.078 −0.178 −0.220 −0.272 −0.334 −0.266 −0.126 0 0.150

0.326 0.248 0.256 0.1 −0.068 −0.262 −0.558 −0.740 −0.899 −0.956 −0.942 −0.928 −0.880 −0.980 −1.040 −0.830

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