# A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer
## 3 Experimental Results and Discussion
The circular fitting of average trajectory produces \(x_{c}\) = 0.00303 mm, \(y_{c}\) = 0.00252 mm and \(R_{c}\) = 8.9873 mm (this radius being very close to the amplitudes of \(x_{a}\) and \(y_{a}\) already revealed in Equation (1)). A first conventional graphical description of the circularity error of the average trajectory (as  
Figure 11: The average evolution \(y_{A\prime}\) of the residual \(y_{r}\) for \(y\)-motion. Figure 10: The evolution of \(x_{A\prime}\) and an analytical model \(x_{A\prime}\) (with \(n\) = 105) in area A of Figure 8.  
a first trajectory fitting residual, TFR\({}_{1}\)) is available in polar coordinates (\(d_{i1}\), \(\alpha_{i1}\)), related to the least square circle, with \(d_{i1}\), \(\alpha_{i1}\) defined as:  
\[d_{i1}=\left|\sqrt{\left(x_{ei}-x_{c}\right)^{2}+\left(y_{ei}-y_{c}\right)^{2}} -R_{c}\right|\alpha_{i1}=\arctan^{4}\left(\frac{y_{ei}-y_{c}}{x_{ei}-x_{c}}\right) \tag{5}\]  
Here \(d_{i1}\) is the distance from average trajectory to the least square circle, \(\alpha_{i1}\) is the polar angle, with \(arctan^{4}\) the inverse of tangent function in four quadrants. This TFR\({}_{1}\) is also available in Cartesian coordinates as a curve described by a movable point having \(d_{i1}\cdot cos(\alpha_{i1})+x_{c}\) as abscissa and \(d_{i1}\cdot sin(\alpha_{i1})+y_{c}\) as ordinate. If the average trajectory is a perfect circle, then TFR\({}_{1}\) is a point placed in the center of the least square circle. Figure 12 presents the TFR\({}_{1}\) of the average trajectory with a circular grid (with a 20 \(\upmu\)m increment on radius). Here the maximum value of distance \(d_{i1}\) is 145.8 \(\upmu\)m.  
It is interesting to explain why TFR\({}_{1}\) from Figure 12 has four almost similar lobes. The dominant component (as amplitude) in \(x_{e}\)-motion is \(x_{a}\), while the dominant component in \(y_{e}\)-motion is \(y_{a}\) (with \(x_{a}\) and \(y_{a}\) experimentally revealed by fitting and depicted in Equation (1) as pure harmonic motions). As previously shown, these two components (having the same amplitude and angular frequency) are not rigorously shifted with \(\pi/2\) (as expected).This means that the dominant part of the average trajectory (generated by \(x_{a}\) and \(y_{a}\) motions composition) is not a circle (as expected) but an ellipse. The circular fitting of an ellipse produces a least square circle which intersects the ellipse in four points and a TFR\({}_{1}\) with four lobes, according to the simulation results from Figure 13 (with an elliptical trajectory generated by two harmonic signals shifted with 2.1863 radians). A better explanation for these four lobes in Figure 12 is produced if over this figure is added the TFR\({}_{1}\) of the trajectory generated only by \(x_{a}\) and \(y_{a}\)-motions, with green color, as Figure 14 indicates (here both curves being traversed counter clockwise). A better approach to the shape of TFR\({}_{1}\) involved in Figure 14 is produced if the distances \(d_{i1}\) from Equation (5) are calculated related by a circle with smaller radius than the radius of the least square circle (\(R_{c}\)), as TFR\({}_{1a}\). As example, in TFR\({}_{1a}\) from Figure 15, this radius is \(R_{c}-0.025\)\(\upmu\)m.  
Figure 12: The evolution of TFR\({}_{1}\). Figure 14: The evolution of \(\mathrm{TFR_{1}}\) generated by \(x_{e}\) and \(y_{e}\) and \(\mathrm{TFR_{1}}\) generated only by \(x_{a}\) and \(y_{a}\) (with green color). Figure 13: The evolution of \(\mathrm{TFR_{1}}\) generated by circular fitting on an elliptical trajectory (simulation). Figure 9:0 The evolutions of \(\mathrm{TFR_{1a}}\). For P&A evaluation of the average circular trajectory (the result of \(x_{e}\) and \(y_{e}\) simultaneous motions), an important item is the size of the surface delimited by the trajectories fitting residuals (TFR\({}_{1}\) and TFR\({}_{1a}\)). Each area is a sum of the areas of \(N\)-Ineighboring triangles. All triangles share a common vertex placed in the origin of coordinate systems. The other two vertices are two successive points on TFR.