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FIGURE 8 Elementary two-dimensional affine transformations. (Left column) Effects produced by off-diagonal elements in the matrix logarithm of a transformation matrix. Balanced off-diagonal elements produce rotations (left center). Unbalanced off-diagonal elements result in shears along one of the two axes (left top and left bottom). Like rotations, elementary shears do not change areas or volumes. (Center column) When all elements in the matrix logarithm of a transformation matrix are zero, an identity transformation is specified. (Right column) Effects produced by diagonal elements in the matrix logarithm of a transformation matrix. Balanced diagonal elements produce global rescaling (right center). Unbalanced diagonal elements result in anisotropic rescaling along the corresponding axis. Global rescaling and anisotropic rescaling both generally change areas or volumes, though these quantities can be preserved if anisotropic rescaling along one axis exactly compensates for rescaling along another axis. These relationships all generalize to three dimensions.

example, the instantaneous elementary matrix for shearing along x is

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