## 2 The One-Shot Projection Moire: Model and Experimental Setup
### The Proposed Moire Model for Inflated Membranes
An elementary segment of the deformed surface is defined in the XYZ global coordinate system by the \"displaced\" points \(P_{1}{{}^{\prime}}(x_{1}{{}^{\prime}},y_{1}{{}^{\prime}},z_{1}{{}^{\prime}})\), \(P_{2}{{}^{\prime}}(x_{2}{{}^{\prime}},y_{2}{{}^{\prime}},z_{2}{{}^{\prime}})\), \(P_{3}{{}^{\prime}}(x_{3}{{}^{\prime}},y_{3}{{}^{\prime}},z_{3}{{}^{\prime}})\) and \(P_{4}{{}^{\prime}}(x_{4}{{}^{\prime}},y_{4}{{}^{\prime}},z_{4}{{}^{\prime}})\). The corresponding undeformed positions are defined by points \(P_{1}(x_{1},y_{1},z_{1})\), \(P_{2}(x_{2},y_{2},z_{2})\), \(P_{3}(x_{3},y_{3},z_{3})\) and \(P_{4}(x_{4},y_{4},z_{4})\). Let us assume that the tested membrane has a planar undeformed configuration, that is \(z_{i}=0\), consistent with the inflation test protocol: the undeformed membrane lies in the reference plane \(\pi\) of moire measurements. Figure 3a shows how the \(P_{1}P_{2}P_{3}P_{4}\) surface portion deforms into the corresponding \(P_{1}{{}^{\prime}}P_{2}{{}^{\prime}}P_{3}{{}^{\prime}}P_{4}{{}^{\prime}}\) surface portion. Let \(R_{\mathrm{c}}\) be the average curvature radius of the deformed surface segment, defined as the average distance of points \(P_{1}{{}^{\prime}}\), \(P_{2}{{}^{\prime}}\), \(P_{3}{{}^{\prime}}\) and \(P_{4}{{}^{\prime}}\) from the center of curvature \(C(x_{\mathrm{c}},y_{\mathrm{c}},z_{\mathrm{c}})\) of the segment itself. If the \(P_{1}{{}^{\prime}}P_{2}{{}^{\prime}}P_{3}{{}^{\prime}}P_{4}{{}^{\prime}}\) surface is locally approximated by a spherical segment (as it may occur for an inflated membrane), it holds:  
\[\left\{ \begin{aligned} \left(x_{1}{{}^{\prime}}-x_{\mathrm{C}} \right)^{2}+\left(y_{1}{{}^{\prime}}-y_{\mathrm{C}}\right)^{2}+\left(z_{1}{{}^ {\prime}}-z_{\mathrm{C}}\right)^{2}&=R_{\mathrm{C}}^{2}\\ \left(x_{2}{{}^{\prime}}-x_{\mathrm{C}}\right)^{2}+\left(y_{2}{{}^{ \prime}}-y_{\mathrm{C}}\right)^{2}+\left(z_{2}{{}^{\prime}}-z_{\mathrm{C}} \right)^{2}&=R_{\mathrm{C}}^{2}\\ \left(x_{3}{{}^{\prime}}-x_{\mathrm{C}}\right)^{2}+\left(y_{3}{{}^ {\prime}}-y_{\mathrm{C}}\right)^{2}+\left(z_{3}{{}^{\prime}}-z_{\mathrm{C}} \right)^{2}&=R_{\mathrm{C}}^{2}\\ \left(x_{4}{{}^{\prime}}-x_{\mathrm{C}}\right)^{2}+\left(y_{4}{{}^ {\prime}}-y_{\mathrm{C}}\right)^{2}+\left(z_{4}{{}^{\prime}}-z_{\mathrm{C}} \right)^{2}&=R_{\mathrm{C}}^{2}\end{aligned} \right. \tag{6}\]  
Equation (6) corresponds to four relationships containing in total 16 unknown parameters: 12 coordinates of \"displaced\" points \(P_{1}{{}^{\prime}}\), \(P_{2}{{}^{\prime}}\), \(P_{3}{{}^{\prime}}\) and \(P_{4}{{}^{\prime}}\) (i.e., \(x_{1}{{}^{\prime}}\), \(y_{1}{{}^{\prime}}\), \(z_{1}{{}^{\prime}}\), \(x_{2}{{}^{\prime}}\), \(y_{2}{{}^{\prime}}\), \(z_{2}{{}^{\prime}}\), \(x_{3}{{}^{\prime}}\), \(y_{3}{{}^{\prime}}\), \(z_{3}{{}^{\prime}}\), \(x_{4}{{}^{\prime}}\), \(y_{4}{{}^{Substituting the first two relationships of Equation (8) into the third relationship of the same equation, and recalling the last relationship of Equation (7), it can be written:  
\[\left\{ \begin{aligned} & U_{T1}=\sqrt{\left({x_{1}}^{\prime}-{x_{1}} \right)^{2}+\left({y_{1}}^{\prime}-{y_{1}}\right)^{2}}={z_{1}}^{\prime}\tan \alpha\\ & U_{T2}=\sqrt{\left({x_{2}}^{\prime}-{x_{2}}\right)^{2}+\left({y _{2}}^{\prime}-{y_{2}}\right)^{2}}={z_{2}}^{\prime}\tan\beta\\ & U_{T3}=\sqrt{\left({x_{3}}^{\prime}-{x_{3}}\right)^{2}+\left({y _{3}}^{\prime}-{y_{3}}\right)^{2}}={z_{3}}^{\prime}\tan\gamma\\ & U_{T4}=\sqrt{\left({x_{4}}^{\prime}-{x_{4}}\right)^{2}+\left({y _{4}}^{\prime}-{y_{4}}\right)^{2}}={z_{4}}^{\prime}\tan\delta\end{aligned} \right. \tag{9}\]  
where \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) are the parallax angles defined in projection moire [31, 35, 36]. As mentioned before, Equation (6) contains 16 unknown parameters: 12 coordinates of \"displaced\" points \(P_{1}\), \(P_{2}\), \(P_{3}\)\({}^{\prime}\) and \(P_{4}\)\({}^{\prime}\), 3 coordinates of curvature center C and the curvature radius of the deformed surface segment.