Computerized determination of gear tooth surface as envelope to two parameter family of surfaces

The contents of this paper covers: (1) Necessary and sufficient conditions of the envelope to a two-parameter family of surfaces. (2) Applications to the generation of the gear tooth surface by a tool surface. (3) Avoidance of singularities and undercutting of the generated tooth surface. (4) Deviations of the real gear tooth surface from the envelope to the family of two-parameter family of tool surfaces. The developed theory is applied and illustrated for the case of generation of a helical gear by a grinding involute worm that is performing motion with two independent parameters.