This study relates to the prediction of the angular positions of supernumerary screenbows and rainbows, in the case of a refractive sphere illuminated by a point source placed at a distance of h from its center; for h→∞, the incident light beam becomes parallel. The screenbow appears on a spherical screen whose center is that of the sphere and which intercepts the tangential caustic surface. The rainbow, specific to the water drop, but here generalized to any refractive sphere, corresponds to a screenbow produced on a "screen" placed at an infinite distance. This paper uses exact graphical representations of the wavefronts associated with rainbows resulting from k internal reflections to illustrate how the angular positions of the supernumerary rainbows and the positions of the corresponding supernumerary bows on screens are to be calculated. All considerations are made within the framework of geometrical optics being, on the one hand, the limit of the electromagnetic theory as the wavelength goes to 0, and, on the other hand, complemented by the Gouy phase shift theory.