Homogeneous deformations: determination by fringe vectors in hologram interferometry.

When an object undergoes a combination of minute rigid-body motion and homogeneous deformation, its vectorial displacements can be described by a general, 3 x 3 matrix transformation. This same matrix can be used to transform the sensitivity vector of hologram interferometry into a fringe vector that defines the fringes as laminae that are intersected by the surface of the object. When fringes are observable on more than one surface of a three-dimensional object whose shape is known, it is possible to determine the fringe vector from the shape and spacing of the fringes on the object. If three holographic views are available, from which three fringe vectors can be determined for three known sensitivity vectors, it is possible to determine the transformation matrix, and this, in turn, can be decomposed into deformation and rotation matrices. More than three views allow the use of least-square-error theory to minimize errors in data taking.