In the current moiré literature, techniques to determine displacements, strains, and techniques to get geometrical parameters of surfaces using the shadow-projection moiré method are considered two separated branches of moiré. We have formulated a mathematical model that shows a deeper commonality between the two moiré applications: strain fields and surfaces are tensors of the second order. A direct consequence of this property is that a system of orthogonal grids is required in both cases when Cartesian tensors are utilized. The two systems of lines projected on a surface to get its contour are assimilated to parametric lines used in differential geometry to describe a surface. The classical moiré equations of projection and observation from infinity are extended to more general conditions of projection and observation. The use of four projectors (i.e., two groups of two projectors) in a mutually orthogonal system with one camera is shown to provide the necessary means to implement the model for high-accuracy contouring. Geometrical primitives are introduced to provide a simple and direct procedure to reduce all the measured values to a preselected coordinate system. Different views of the same surface are merged to the selected coordinate system directly without the need to introduce markers on the surface or utilize correlation methods to identify identical regions. Examples of the application of the new model of contouring to practical cases are presented in the companion paper "A general model for moiré contouring, part 2: applications

### A general model for moiré contouring. Part I: Theory

#### Abstract

In the current moiré literature, techniques to determine displacements, strains, and techniques to get geometrical parameters of surfaces using the shadow-projection moiré method are considered two separated branches of moiré. We have formulated a mathematical model that shows a deeper commonality between the two moiré applications: strain fields and surfaces are tensors of the second order. A direct consequence of this property is that a system of orthogonal grids is required in both cases when Cartesian tensors are utilized. The two systems of lines projected on a surface to get its contour are assimilated to parametric lines used in differential geometry to describe a surface. The classical moiré equations of projection and observation from infinity are extended to more general conditions of projection and observation. The use of four projectors (i.e., two groups of two projectors) in a mutually orthogonal system with one camera is shown to provide the necessary means to implement the model for high-accuracy contouring. Geometrical primitives are introduced to provide a simple and direct procedure to reduce all the measured values to a preselected coordinate system. Different views of the same surface are merged to the selected coordinate system directly without the need to introduce markers on the surface or utilize correlation methods to identify identical regions. Examples of the application of the new model of contouring to practical cases are presented in the companion paper "A general model for moiré contouring, part 2: applications
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2008
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11589/9282`
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