## 2 The One-Shot Projection Moire: Model and Experimental Setup
### The Proposed Moire Model for Inflated Membranes
\tag{10}\]  
By combining Equations (6), (9) and (10), it is hence possible to determine the remainingnot displace in the transverse direction to the selected diameter. Boundary conditions taking into account the fixture applied to the membrane boundary also may be considered. For example, the displacements of the circular ring corresponding to the constrained boundary of the membrane can be set equal to 0: the conditions \(x^{\prime}=x,y^{\prime}=y\) and \(z^{\prime}=z\) hold true for the points of surface elements that include some constrained edges. An alternative strategy for estimating coordinates may regard points close to the pole of the inflated membrane: the distance \(\Delta\xi_{i}=(\xi_{i}-\xi_{\rm o})\) between a generic point \(P_{i}\) of the currently selected diameter and the membrane pole \(P_{\rm o}\) (i.e., the center point of inflated membrane, which should displace only in thez-direction but not in the x/ydirections) is multiplied by the nominal strain \(\varepsilon_{\rm p}\) generated by the applied pressure. Nominal strain may even be corrected to \(\varepsilon_{\rm p}\)corr to account for the local variations of membrane thickness, for example, by using in first approximation the \(\varepsilon_{\rm p}\)corr/\(\varepsilon_{\rm p}=\mathsf{t}_{\rm o}/\mathsf{t}_{\rm loc}\) relationship taken from linear elasticity where \(\mathsf{t}_{\rm loc}\) and \(\mathsf{t}_{\rm o}\), respectively, are the local and nominal thickness values. The displaced position \(\xi_{i^{\prime}}\) of point \(P_{i}^{\prime}\) is determined from the \(\Delta\xi_{i^{\prime}}=(1+\varepsilon_{\rm p})\xi_{i}\) equation, where the distance \(\Delta\xi_{i^{\prime}}=(\xi_{i}^{\prime}-\xi_{\rm o})\) is referred to the deformed configuration. The local \(\xi^{\prime}\)-coordinates are finally transformed into x\({}^{\prime}\)- and y\({}^{\prime}\)-coordinates of deformed points based on the orientation of the selected diameter, and the corresponding in-plane displacements are determined from Equation (6). Interestingly, [49] analyzed the strain distributions in the radial (longitude) and circumferential (latitude) directions that develop in an inflated natural rubber membrane of diameter 120 mm and thickness 0.9 mm, very similar to the test case documented in the present article. Surface elements located within a circular area with a radius up to 20% of the membrane's radius can be considered in an equi-biaxial state. Furthermore, radial strain remains nearly constant over the whole membrane. This information may bevery useful in the above-mentioned alternative strategy for determining the \(x^{\prime}\), \(y_{1}^{\prime}\), \(x_{2}^{\prime}\), \(y_{2}^{\prime}\), \(x_{3}^{\prime}\), \(y_{3}^{\prime}\), \(x_{4}^{\prime}\), \(y_{4}^{\prime}\) coordinates of the displaced points. Although the process described above may appear complicated, it should be noted that points of adjacent surface segments obviously share the same coordinates and the same displacements. This reduces significantly the number of unknowns and speeds up the contouring process. However, continuity conditions on curvature radii and surface slopes at the interface points between adjacent elements also have to be considered. Information on local slopes and curvatures at each point of the deformed surface can be obtained by differentiating the moire patterns relative to the two systems of orthogonal lines projected onto the surface. Slopes of X-Z profiles are proportional to first-order spatial derivatives \(\partial w/\partial x\) of moire patterns, while slopes of Y-Z profiles are proportional to first-order spatial derivatives \(\partial w/\partial y\). Curvature terms are proportional to second-order spatial derivatives \(\partial^{2}w/\partial x^{2}\), \(\partial^{2}w/\partial y^{2}\) and \(\partial^{2}w/\partial x\partial y\) of moire patterns. Values of coordinates \(x\) and \(y\) at which first- and second-order derivatives are evaluated obviously are those relative to the deformed configuration.