## 2 The One-Shot Projection Moire: Model and Experimental Setup
### The Proposed Moire Model for Inflated Membranes
\[\Delta\ell^{\prime}=R_{c}\Delta\phi \tag{2}\]  
where \(R_{c}\) is the local curvature radius of the deformed membrane; the angle \(\Delta\phi\) is limited by the arc \(\widehat{P_{1}{}^{\prime}B_{2}{}^{\prime}}\), portraying the local surface's deformation. If the deformed arc belongs to a circumference, the curvature radius is constant. Otherwise, the average of the curvature radii evaluated at points \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) has to be determined. For example, for the X-Z deformed profile of Figure 2c, the values of curvature radii at \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) are obtained by evaluating the \(1/(\partial^{2}w/\partial x^{2})\) function at \({x_{\mathrm{P1}}}^{\prime}\) and \({x_{\mathrm{P2}}}^{\prime}\). The same argument can be made for the Y-Z deformed profile, considering the \(1/(\partial^{2}w/\partial y^{2})\) function and coordinates \({y_{\mathrm{P1}}}^{\prime}\) and \({y_{\mathrm{P2}}}^{\prime}\). The \(w\)-displacements are determined with projection moire. Referring to the scheme of Figure 2, the initial length \(\Delta\ell\) of the undeformed configuration \(\widehat{P_{1}P_{2}}\) is:  
\[\Delta\ell=(R_{c}-z_{1})\tan\Delta\phi \tag{3}\]  
The length \(\Delta\ell^{\prime}\) of the deformed configuration \(\widehat{P_{1}{}^{\prime}P_{2}{}^{\prime}}\) can be determined because X positions of points \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) are known from the image (for example, by multiplying the pixel difference between the currently analyzed position and the origin of the reference system by the pixel size), and the \(w\)-displacements \(z_{1}\) and \(z_{2}\) are known from projection moire measurements. By differentiating moire patterns with respect to \(\mathbf{X}\), the radius of curvature \(R_{c}\) of the deformed surface also can be determined. By combining Equations (2) and (3), it is possible to compute \(\Delta\ell\) and \(\Delta\phi\). Finally, the \(u\)-displacement of point \(P_{2}\) (i.e., \(u_{2}\)) is the difference between the X-coordinate of point \({P_{2}}^{\prime}\) and the length \(\Delta\ell\). The same arguments can be made for the Y-Z profiles. It can be seen from Figure 3 that it holds:  
\[u_{2}=z_{2}\tan\Delta\phi \tag{4}\]  
Figure 3 also shows that the \(\Delta\phi\) angle corresponds to the slope of the deformed arc \(\widehat{P_{1}{}^{\prime}P_{2}{}^{\prime}}\) at point \({P_{2}}^{\prime}\) because the local curvature radius \(\overline{CP_{2}}\) is normal to \(\widehat{P_{1}{}^{\prime}P_{2}{}^{\prime}}\). Hence, from geometrical considerations, Equation (4) can be rewritten as:  
\[u_{2}=z_{2}\left.\frac{\partial w}{\partial x}\right|_{x=x_{2}} \tag{5}\]  
In summary, the \(u\)-displacement of point \(P_{2}\) can be directly determined from the moire fringe pattern relative to \(w\)-displacements and the corresponding derivative \(\partial w/\partial x\) if the  
Figure 3: Determination of 3D displacement field for a generically deformed membrane subjected to inflation: (**a**) deformed configuration vs. undeformed configuration; and (**b**) components of displacement vector for a generic point of the membrane with indication of azimuthal and parallax angles. deformed configuration coordinate \(x_{2}{{}^{\prime}}\) is known. A similar relationship can be found for the Y-Z profiles. This property may be very useful for estimating in-plane displacements from the available \(w\)-displacement moire pattern. It should be noted that Equation (3) loses its validity if the undeformed specimen is curved. In fact, the curvature radius of element \(\overline{P_{1}P_{2}}\) changes from \(R_{\mathrm{c}}{{}^{\mathrm{o}}}\) (initial configuration) to \(R_{\mathrm{c}}\) (deformed configuration \(\widehat{P_{1}P_{2}}\)). The length variation from \(\Delta\ell\) to \(\Delta\ell^{\prime}\) has to be determined with shells theory [47; 48]. Curvature information is still extracted from moire data [35; 36]. The simple derivation described above may be generalized to the 3D case (Figure 3).