Based on a recently developed geometric approach to the theory of gearing that does not make use of any reference systems, this paper presents some useful relations between the geometric properties of the enveloping surface and those of its envelope. Treating vectors as such, that is without expressing their components in any reference systems, it is possible to obtain compact expressions for the coefficients of the first and second fundamental forms of the envelope surface. These coefficients show to be central in the determination of the contact matrix between mating surfaces. Moreover, since this approach is coordinate free, it is valid regardless of the reference frame actually employed to perform calculations and allows a, hopefully, clearer understanding of the roles played by the intrinsic geometric properties of the enveloping surface, the relative position of the gear axes and the gear ratio.
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