  We make use of point transformations to introduce new canonical variables for
systems defined on a finite interval and on the half-line so that new position
variables should take all real values from $-\infty$ to $\infty$. The
completeness of eigenvectors of new momentum operators enables us to formulate
time sliced path integrals for such systems. Short time kernels thus obtained
require extension of the range of variables to the covering space in order to
take all reflected paths into account. Upon this extension we determine phase
factors attached to the amplitude for paths reflected at boundaries by taking
singularities of the potential into account. It will be shown that the phase
factor depends on parameters that characterize the potential; and further that
the well-know minus sign in the amplitude for odd times reflection of a
particle in a box should be understood as the special case for the
corresponding value of the parameter of the potential.
