  According to a first-principles approach, we clarify electric cloaks
universality concerning a new class of phase transitions between negative
pathway (NGP) and normal pathway of electric displacement fields, which are
driven by the geometric shape of the cloak. We report that the NGP arises from
shape-enhanced strong negative electric polarization, and that it is related to
a symmetric oscillation of the paired electric permittivities, which are shown
to satisfy a sum rule. The NGP does not occur for a spherical cloak, but
appears up to maximum as the ratio a/b between the long and short principal
axis of the spheroidal cloak is about 5/2, and eventually disappears as a/b
becomes large enough corresponding to a rod-like shape. Then, the cloaking
efficiency is compared between different geometrical shapes. The possibility of
experiments is discussed. This work has relevance to crucial control of
electric fields and to general physics of phase transitions.
