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Translations can then be placed in the fourth column of the matrix to reposition the origin to any desired location. If it is necessary to ensure that the axis of rotation passes through some particular point, the first step is to compute all eigenvalues and eigenvectors (including the imaginary ones) of the preceding matrix. If these eigenvectors are used as columns of a matrix V, the eigenvalues can be placed along the diagonal of another matrix D so that V*D*V—1 is equal to the original matrix. One of the eigenvectors will be the vector and the corresponding column of V should be replaced by a vector that represents a point on the desired axis of rotation, keeping the fourth element as unity. The desired matrix can then be computed as V * D * V—1.

FIGURE 7 Illustration of the numerical example of two-dimensional affine transformation. The light gray axes are centered on the coordinate (11.8514, 3.0593), which is unchanged by the transformation. These axes are initially oriented 54.9232° counterclockwise to the coordinate axes. The image is rescaled anisotropically along the gray axes, and then the image and the gray axes are rotated 24.1889° clockwise. Note that within the frame of reference defined by the gray axes, the transformation only involves anisotropic rescaling.

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