Efficient Least Squares Coordinate Transformation for an Arbitrary Number of Parameters

By means of constructing equivalent linear problems, Greenfeld (1997) derived closed-form formulas for four- and six-parameter least squares coordinate transformations between two planar systems, so that users can obtain direct solutions on spreadsheets. However, the three- and five-parameter cases cannot be converted to equivalent linear problems for explicit solutions. An alternative approach is developed here to fully utilize the power inside spreadsheets and solve transformation problems more directly and efficiently, without reformulating the problems for linearity. The new approach is insensitive to the number of parameters, and handles the transformation problems in a unified way. It offers not only the full rigor of least squares, but also the simplicity of being completely spreadsheet based, and eliminates the usual linearization and programming requirements on the part of the user. The new method is further extended to three-dimensional conformal transformations, while practical hints on obtaining provisional solutions and fine-tuning solver parameters to avoid numerical hazards are also discussed.