Euler–Savary formula for the homothetic motion in the complex plane C

This brief note expanding on one aspect of paper [1], which deals with the complex number form of Euler–Savary formula. This formula is the relationship between the radius of the curvature and the curvature center of a path drawn by a point in the coplanar rolling of two conjugate curves on one another. We study complex number form of the Euler–Savary formula for one parameter homothetic planar motion and the case of the homothetic scale h is equal to 1, given in [1], will appear as a special case of this study