## 2 The One-Shot Projection Moire: Model and Experimental Setup
### The Proposed Moire Model for Inflated Membranes  
The IMPM setup recalled in Section 2.1 brings three important questions: (i) How many systems of lines should be projected onto a surface in order to determine the whole 3D deformation of that surface? (ii) Should those systems of lines have different directions? (iii) Can in-plane displacements of a deformed object be measured without physically  
Figure 1: (a) IMPM optical setup of [34] combining intrinsic and projection moire; (b) displacement determination from moiré pattern. (taken from [34]). applying a grating onto the object surface? To answer these questions, the present study proposed a one-shot projection moire optical setup for reconstructing the 3D deformed configuration of the loaded object. A simple analytical model that evaluates the deformed object shape starting from the curvature information was developed and integrated with the projection moire data. The topography of an object's surface is fully defined if the coordinates of points sampling the surface are known. The displacement vector of a given point \(P(x,y,z)\) of the object's surface that moves to the position \(P^{\prime}(x^{\prime},y^{\prime},z^{\prime})\) in the deformed configuration is defined by the differences of coordinates \(\{(x^{\prime}-x);(y^{\prime}-y);(z^{\prime}-z)\}\) between \(P^{\prime}\) and \(P\). Projection moire can give information on the Z-coordinates of surface points as well as on slopes and curvatures of the surface, provided that cross-gratings are projected onto the specimen [31; 35; 36]. The modulation of vertical lines projected onto the specimen gives information on the X-Z surface profiles defined by coordinates \(x\) and \(z\), while the modulation of projected horizontal lines provides information on the Y-Z surface profiles defined by coordinates \(y\) and \(z\). The X-Z and Y-Z profiles obviously share the same \(z\) at their intersections. By combining such information, it is possible to determine the position of each surface point with respect to the reference system selected in the measurements. Each surface is a tensor entity with two curvatures that refer to orthogonal directions, as well as a torsional curvature. This model resembles the deformation of a solid body: the surface strain tensor includes stretching/compression/shear components (see [31; 35; 36] for detailed explanations of the connections between moire and differential geometry). The surface can be reconstructed with respect to a properly selected reference system if principal curvatures are known, similar to recovering a strain field from principal strains. In order to prove the above statements, the schematic of Figure 2 can be considered. A circular membrane is inflated by pressure, thus passing from the undeformed configuration (Figure 2a) to the deformed configuration (Figure 2b). The simplest case regards an axiymmetrically deformed specimen (i.e., there are only two displacement components \(u=v\) and \(w\), respectively, in the X/Y- and Z-directions; see Figure 2b) and the reference configuration is a flat surface (see Figure 2a). Figure 2c presents a X-Z diametrical section of the deformed membrane. The undeformed state corresponds to the straight segment \(\overline{P_{1}P_{2}}\) while the deformed configuration corresponds to the arc \(\overline{P_{1}P_{2}}\). Furthermore, if point \(P_{1}\) lies on the symmetry axis of the investigated object, it does not displace in the X/Ydirection. Under the action of applied loads, point \(P_{1}\) in the reference (undeformed) state displaces to the deformed position of point \(P_{1}\), while point \(P_{2}\) displaces to point \(P_{2}\). The length of the deformed arc \(\overline{P_{1}P_{2}}\) is:  
Figure 2: Determination of the 3D displacement field for an axiallysymmetric deformed membrane subjected to inflation: (**a**) undeformed configuration; (**b**) deformed configuration; and (**c**) deformed profile in the diametrical section X–Z of the inflated membrane.