Form tolerances are given by ISO in terms of the minimum separation zone which is a concept based upon the Chebyshev criterion. But this concept is only recommended without any specific methods to be fulfilled, and as a result, form tolerances are being evaluated by means of diverse methods and algorithms. In spite of this method divergence, most of the commercial coordinate measuring equipments adopt the least squares method based upon the Gaussian criterion because of it's simplicity. The least squares method could be implemented in a one-step procedure for linear models while most of the other methods require iterative numerical procedures which could suffer in time consuming computations or local minimum problems. The error to be minimized by the least squares should be defined along the normal direction with respect to the ideal-reference of the object which is to be measured, otherwise the relationships between the parameters of the ideal-reference would be inconsistent according to the orientations of error definition.
The least squares method based upon this normal direction error has already been suggested since the 1980s, but it was only to be applied to straightness and restricted flatness evaluations. This dissertation proposes a new algorithm of normal least squares of which evaluation area contains circles and spheres with good approximations as well as lines and planes. In this algorithm the least squares problem could be solved by formulating an eigenvalue problem for the variance-covariance matrix. Some comparisons with the existing similar algorithms are presented for the cases of lines and planes. For the cases of circles and spheres, the algorithm could be theoretically used only for a specific type of data, but the results of computer simulations show the robustness and goodness of this algorithm compared to the existing relevant ones.
The following characteristics of the new algorithm could be proved by some computer simulations; when used for flatness evaluation the algorithm could be fulfilled overcoming the restrictions of the existing algorithms, and when used for circularity evaluation there was no need for the sample data to be well aligned with respect to the measuring machine's coordinate origin nor the data to be sampled along the whole circumference.
