Determination of envelope to family of planar parametric curves and envelope singularities

Abstract A parametric family Σ φ of planar curves generated by the tool profile Σ r is considered. The generated profile Σ p of the workpiece (gear) is the envelope to the family Σ φ . The authors propose simplified approaches for determination of the envelope, its singularities and conditions of existence of a point of regression. The proposed developments are illustrated with the design and generation of Root's blowers.