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Abstract.The relationships between various numerical methods for obtaining polynomial approximations to the first derivative of a known function are investigated, and their computational advantages discussed. Optimum sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error, and geometric sequences, though non-optimal in this sense, are considered for computational reasons.

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Seven-parameter conformal coordinate transformations, also known as Helmert transformations, can be constructed in more than one way. Two possible orderings of the rotations are in common use, giving rise to Helmert versions 1 and 2. It is demonstrated how the rotation parameters of either version can be converted into the rotation parameters of the other. This is useful when software is designed for the other version. It also enables computation of the same-formula inverse transformation by changing the sign of the equivalent 'other version' parameters. These results were primarily intended for conformal transformations between geodetic datums. They can, however, be extended to coordinate transformations in disciplines such as photogrammetry where rotations sometimes exceed 90 degrees.

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