Techniques to analyze displacements and contouring of surfaces

In the current literature techniques to determine displacements, strains and geometrical parameters of surfaces are looked upon as two different branches of the application of tools developed basically by researchers working in the field of Experimental Mechanics. This paper formulates a mathematical model of the contouring techniques that shows a deeper commonality of the basic mathematics needed to analyze surfaces and the mathematics utilized in the analysis of strain fields. This mathematical model is based on the differential geometry analysis of surfaces. Surfaces in 3-D that can be described by a regular parametric representation of the type X=X(u,v) , where X is a vector in 3-D that is a vectorial function of parameters u, v; these surfaces are called simple surfaces. The parameters u,v, establish a mapping of a patch of a plane into a surface S. Most of the surfaces utilized in engineering practice fall within the category of simple surfaces. A basic theorem of the theory of simple surfaces implies that simple surfaces can be represented by tensors of the second order, consequently the theory of strain and the theory of simple surfaces have a common mathematical structure. This common mathematical structure implies that the same procedures needed to analyze strains need to be applied to the analysis of surfaces. An immediate consequence of the above reached conclusion is that to study surfaces using Cartesian coordinate systems a set of orthogonal gratings are required. This requirement is unlike most of the presently developed techniques in contouring which utilize one single grating. Many other practical consequences derive from this common mathematical framework which are extremely useful for engineering applications