  By investigating wave properties at cloak boundaries, invisibility cloaks
with arbitrary shape constructed by general coordinate transformations are
confirmed to be perfectly invisible to the external incident wave. The
differences between line transformed cloaks and point transformed cloaks are
discussed. The fields in the cloak medium are found analytically to be related
to the fields in the original space via coordinate transformation functions. At
the exterior boundary of the cloak, it is shown that no reflection is excited
even though the permittivity and permeability do not always have a perfect
matched layer form. While at the inner boundary, no reflection is excited
either, and in particular no field can penetrate into the cloaked region.
However, for the inner boundary of any line transformed cloak, the permittivity
and permeability in a specific tangential direction are always required to be
infinitely large. Furthermore, the field discontinuity at the inner boundary
always exists; the surface current is induced to make this discontinuity
self-consistent. For a point transformed cloak, it does not experience such
problems. The tangential fields at the inner boundary are all zero, implying no
field discontinuity exists
