Abstract
This paper describes a fundamental geometric relationship between pitch curves and tooth profiles in the spur gearing with variable ratio. The author derives a simple algebraic form of the extended Euler-Savary's equation, which describes a relationship between radii of pitch curves and radii of curvatures of tooth profiles in this gearing, with elementary geometry. This formula is based on the Willard Gibbs' dissertation. Here, a slope of common tangent of pitch curves at the pitch point to the center line of gearing is mainly considered independently of curvatures of pitch curves. Further, the author derives an analogical formula considering a role of the curvature of pitch curves and indicates that it yields to the ordinary Euler-Savary's equation in spur gearing with constant ratio. Finally, an application of these formulae to elliptical gearing is presented.
