  Investigating the geometric effects resulting from the detailed behaviors of
the confining potential, we consider square and circular confinements to
constrain a particle to a space curve. We find a torsion-induced geometric
potential and a curvature-induced geometric momentum just in the square case,
while a geometric gauge potential solely in the circular case. In the presence
of electromagnetic field, a geometrically induced magnetic moment couples with
magnetic field as an induced Zeeman coupling only for the circular confinement,
also. As spin-orbit interaction is considered, we find some additional terms
for the spin-orbit coupling, which are induced not only by torsion, but also
curvature. Moreover, in the circular case, the spin also couples with an
intrinsic angular momentum, which describes the azimuthal motions mapped on the
space curve. As an important conclusion for the thin-layer quantization
approach, some substantial geometric effects result from the confinement
boundaries. Finally, these results are proved on a helical wire.
