  We give some details about the periodic cylindrical solution found by Zhang
and Ou-Yang in [Phys. Rev. E 53, 4206(1996)] for the general shape equation of
vesicle. Three different kinds of periodic cylindrical surfaces and a special
closed cylindrical surface are obtained. Using the elliptic functions contained
in \emph{mathematic}, we find that this periodic shape has the minimal total
energy for one period when the period-amplitude ratio $\beta\simeq1.477$, and
point out that it is a discontinuous deformation between plane and this
periodic shape. Our results also are suitable for DNA and multi-walled carbon
nanotubes (MWNTs)
