## 2 The One-Shot Projection Moire: Model and Experimental Setup
### The Proposed Moire Model for Inflated Membranes
Coordinates \(x_{1}\)\({}^{\prime}\), \(y_{1}\)\({}^{\prime}\), \(x_{2}\)\({}^{\prime}\), \(y_{2}\)\({}^{\prime}\), \(x_{3}\)\({}^{\prime}\), \(y_{3}\)\({}^{\prime}\), \(x_{4}\)\({}^{\prime}\), and \(y_{4}\)\({}^{\prime}\) of deformed points can be obtained from Equation (8) once moire fringes are processed and displacements \(u\) and \(v\) are determined; hence, the total number of unknowns reduces to 8, i.e., \(z_{1}\), \(z_{2}\)\({}^{\prime}\), \(z_{3}\)\({}^{\prime}\), \(z_{4}\)\({}^{\prime}\), \(x_{c}\), \(y_{c}\), \(z_{c}\) and \(R_{c}\). Alternatively, one can use an approximated 2D version of Equation (5) to estimate in-plane displacements \(u_{i}\) and \(v_{i}\) (\(i=1,2,3,4\)), respectively, as (\(z_{i}\)\({}^{\prime\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)), where \(z_{i}\)\({}^{\prime\prime}\) are the deformed surface heights (i.e., the red etched lines in Figure 3a limited by \(P_{1}\) and \(Z_{1}\)\({}^{\prime\prime}\), \(P_{2}\) and \(Z_{2}\)\({}^{\prime\prime}\), \(P_{3}\) and \(Z_{3}\)\({}^{\prime\prime}\), \(P_{4}\) and \(Z_{4}\)\({}^{\prime\prime}\)) obtained from the moire pattern for \"undeformed\" points \(P_{1}\), \(P_{2}\), \(P_{3}\) and \(P_{4}\). However, the determination of surface heights \(z_{1}\)\({}^{\prime}\), \(z_{2}\)\({}^{\prime}\), \(z_{3}\)\({}^{\prime}\), \(z_{4}\)\({}^{\prime}\) is not straightforward even if \(u\)- and \(v\)-displacements are known/estimated. In fact, since any location (\(x_{i}\),\(y_{i}\)) of the undeformed configuration displaces to the corresponding location (\(x_{i}\),\(y_{i}\)), each height \(z_{i}\)\({}^{\prime}\) (i.e., the \(w\)-displacement of the \(P_{i}\) point that becomes the \(P_{i}\) point of the deformed surface) has to be referred to the corresponding deformed coordinates (\(x_{i}\), \(y_{i}\)), not to the original coordinates (\(x_{i}\),\(y_{i}\)). Equation (9) relates the total in-plane displacements with the out-of-plane displacements of the membrane segment vertices \(P_{1}\)\({}^{\prime}\), \(P_{2}\)\({}^{\prime}\), \(P_{3}\)\({}^{\prime}\) and \(P_{4}\)\({}^{\prime}\) but introduces four additional unknowns, the parallax angles \(\alpha\) (between \(\overline{CP_{1}}\)\({}^{\prime}\) and \(z_{1}\)\({}^{\prime}\)), \(\beta\) (between \(\overline{CP_{2}}\)\({}^{\prime}\) and \(z_{2}\)\({}^{\prime}\)), \(\gamma\) (between \(\overline{CP_{3}}\)\({}^{\prime}\) and \(z_{3}\)\({}^{\prime}\)) and \(\delta\) (between \(\overline{CP_{4}}\)\({}^{\prime}\) and \(z_{4}\)\({}^{\prime}\)). In this way, the number of unknowns rises to 12 (i.e., \(z_{1}\)\({}^{\prime}\), \(z_{2}\)\({}^{\prime}\), \(z_{3}\)\({}^{\prime}\), \(z_{4}\)\({}^{\prime}\); \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\); and \(x_{c}\), \(y_{c}\), \(z_{c}\) and \(R_{c}\)) but Equations (8) and (9) provide only eight relationships. However, the curvature radii \(\overline{CP_{1}}\)\({}^{\prime}\), \(\overline{CP_{2}}\)\({}^{\prime}\), \(\overline{CP_{3}}\)\({}^{\prime}\) and \(\overline{CP_{4}}\)\({}^{\prime}\) also pass through points \(P_{1}\), \(P_{2}\), \(P_{3}\) and \(P_{4}\) of the undeformed surface. This leads us to write four other relationships (it obviously holds that \(z_{1}=z_{2}=z_{3}=z_{4}=0\)):  
\[\left\{ \begin{aligned} &\sqrt{\left({x_{1}}-{x_{c}}\right)^{2}+\left({y_{1}}-{y_{c }}\right)^{2}+\left({0}-{z_{C}}\right)^{2}}={R_{C}}-\frac{{z_{1}}^{\prime}}{ \cos{\alpha}}\\ &\sqrt{\left({x_{2}}-{x_{c}}\right)^{2}+\left({y_{2}}-{y_{c}} \right)^{2}+\left({0}-{z_{C}}\right)^{2}}={R_{C}}-\frac{{z_{2}}^{\prime}}{ \cos{\beta}}\\ &\sqrt{\left({x_{3}}-{x_{c}}\right)^{2}+\left({y_{3}}-{y_{c}} \right)^{2}+\left({0}-{z_{C}}\right)^{2}}={R_{C}}-\frac{{z_{3}}^{\prime}}{ \cos{\gamma}}\\ &\sqrt{\left({x_{4}}-{x_{c}}\right)^{2}+\left({y_{4}}-{y_{c}} \right)^{2}+\left({0}-{z_{C}}\right)^{2}}={R_{C}}-\frac{{z_{4}}^{\prime}}{ \cos{\delta}}\end{aligned} \right.