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The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917; also called a seven-parameter transformation) is a transformation method within a three-dimensional space. It is frequently used in geodesy to produce distortion-free transformations from one datum to another using: : where * ''X''''T'' is the transformed vector * ''X'' is the initial vector The parameters are: * — translation vector. Contains the three translations along the coordinate axes * — scale factor, which is unitless, and as it is usually expressed in ppm, it must be divided by 1,000,000. * — rotation matrix. Consists of three axes (small rotations around the coordinate axes) , , . The rotation matrix is an orthogonal matrix. The rotation is given in radians. Thus, the Helmert transformation is a similarity transformation. == Calculating the parameters == If the transformation parameters are unknown, they can be calculated with reference points (that is, points whose coordinates are known before and after the transformation. Since a total of seven parameters (three translations, one scale, three rotations) have to be determined, at least two points and one coordinate of a third point (for example, the Z-coordinate) must be known. This gives a system of linear equations with seven equations and seven unknowns, which can be solved. In practice, it is best to use more points. Through this correspondence, more accuracy is obtained, and a statistical assessment of the results becomes possible. In this case, the calculation is adjusted with the Gaussian least squares method. A numerical value for the accuracy of the transformation parameters is obtained by calculating the values at the reference points, and weighting the results relative to the centroid of the points. While the method is mathematically rigorous, it is entirely dependent on the rigour of the parameters that are used. In practice, these parameters are computed from the inclusion of at least three known points in the networks. However the accuracy of these will affect the following transformation parameters, as these points will contain observation errors. Therefore a "real-world" transformation will only be a best estimate and should contain a statistical measure of its quality. It is not always necessary to use the seven parameter transformation, sometimes it is sufficient to use the five parameter transformation, composed of three translations, one rotation (about the Z-axis) and one change of scale. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Helmert transformation」の詳細全文を読む スポンサード リンク
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