Methods are developed for analytically reconstructing the absolute geometric deviations of the running surfaces of all teeth on a gear or pinion from perfect equispaced involute surfaces. Two of the methods utilize a rectangular array of tooth spacing deviation point measurements optimally located at the zeros of Legendre polynomials in the axial and radial directions where profile and lead deviation measurements are assumed to have been made. Each of these two methods utilizes the multiple tooth spacing deviation measurements to determine the absolute spacing deviations of the mean planes of the deviations of the tooth running surfaces determined from profile and lead measurements. A convergent two-dimensional Legendre polynomial representation of tooth surface deviations that interpolates between point measurements of tooth surface deviations located in rectangular arrays at the aforementioned zeros of Legendre polynomials is described. A discussion of the general relationships between adjacent and accumulated tooth spacing deviations is provided. The noninsignificant measurement errors that typically arise in profile, lead, and tooth spacing deviation measurements are taken into account at each step in the work.

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