## 2 Materials and Methods
### Image Post-Processing and Metrics of Performance  
Post-processing software, property of Cornilleau(c), was employed to determine the ball's complex trajectory from the obtained images. The non-orthogonal markers (Figure 2c) were recognized on the traces so that the 2D trajectory could be reconstructed [23; 24]. Six kinematic parameters were finally monitored. They are displayed in Figure 3b. Three parameters were related to the incident trajectory:  
* \(\alpha_{i}\) is the incident angle, in degrees, calculated from the normal to the target surface \(\overrightarrow{n}\).
* \(\|\overrightarrow{V_{i}}\|\) (also denoted \(V_{i}\) in the following) corresponds to the magnitude, in meter per second (m/s), of the incident linear velocity. Considering the experimental setup and coordinate system defined in Figure 3b, \(\overrightarrow{V_{i}}\) had components along the in-plane y- and z-axes,
* \(\|\overrightarrow{\omega_{i}}\|\) (also denoted \(\omega_{i}\) in the following) corresponds to the magnitude, in revolution per second (rev/s), of the incident angular velocity (spin). Here, \(\overrightarrow{\omega_{i}}\) was collinear with the out of plane _x-axes_. Three parameters fully defined the reflected trajectory:
* \(\theta\) is the deviation angle in degrees, calculated from the incident orientation. Clockwise angles were set positive.
* \(\|\overrightarrow{V_{R}}\|\) corresponds to the magnitude, in meter per second (m/s), of the reflected linear velocity. \(\overrightarrow{V_{R}}\) had components along the in-plane y- and z-axes,
* \(\|\overrightarrow{\omega_{R}}\|\) corresponds to the magnitude, in revolution per second (rev/s), of the reflected angular velocity (spin). \(\overrightarrow{\omega_{R}}\) was collinear with the out of plane x-axes,  
Two assumptions accompanied the previous definitions. First, the ball was supposed to remain in the horizontal (xy) plane, suggesting that the effect of gravity was negligible. It was confirmed by verifying (using the circle fit functionality in the image processing software image), that the areas of the ball traces remained constant in one image. Indeed, if the camera were fixed on top of the set-up, the gravity would have contributed to out of plan motion and resulted in smaller traces of the ball on the image. Second, both the incident and reflected trajectories were assumed straight (at least between two consecutive traces), neglecting potential aerodynamics effects.  
Figure 3: **(a)** Raw image of an impact test showing traces of the ball before and after contact. **(b)** Schematization of the kinematic parameters describing the motion of the table tennis ball prior (incident) and after (reflected) impact.  
With the trajectories fully defined, quantitative metrics were set in order to quantify the performance of the impacted polymeric layers. In the previous study focusing on normal impact with no spin [22], the sole linear coefficient of restitution was considered, as defined below:  
\[\text{CR}_{\text{L}}=\frac{\|\overset{\rightarrow}{\
abla_{\text{R}}}\|}{\| \overset{\rightarrow}{\
abla_{\text{i}}}\|}, \tag{1}\]  
\(\text{CR}_{\text{L}}\) is unitless, positive, smaller than 1 (when the target is fixed) and related to the speed performance. \(\text{CR}_{\text{L}}\) remains a relevant metric and was used in this study. The broader spectrum of impact conditions explored suggested the use of two other metrics, namely the deviation angle \(\theta\) and the (unitless) spin ratio SR defined as:  
\[\text{SR}=\frac{\overset{\rightarrow}{\omega}_{\text{R}}}{\overset{\omega} {\omega}_{\text{i}}}, \tag{2}\]  
As mentioned above, both incident and reflected spin axes were collinear with the out of plane _x-axis_. Thus, the spin ratio SR is a real number with absolute value that can be greater than 1. The spin ratio was also studied by [PERSON] et al. [25] in the case of tennis balls. Thus, the aforementioned three parameters were related to the ball's trajectory and aimed at characterizing the performance of the impact between a given _[Ball - Target]_ combination.