  A Bertrand (respectively, Mannheim) curve is a space curve whose principal
normal line is the same as the principal normal (respectively, bi-normal) line
of another curve. By definition, another curve is a parallel curve with respect
to the direction of the principal normal vector. In this paper, we consider the
other cases, that is, a space curve whose tangent (or, principal normal,
bi-normal) line is the same as the tangent (or, principal normal, bi-normal)
line of another curve, respectively. We say that a Bertrand type curve if there
exists such another curve. We clarify that the existence conditions of Bertrand
type curves in all cases. There are times when the Bertrand type curve does not
exist. On the other hand, since the another curve may have singular points, we
also consider curves with singular points. As smooth curves with singular
points, it is useful to use the framed curves in the Euclidean space. Then we
define and investigate Bertrand framed curves. We also clarify that the
existence conditions of the Bertrand framed curves in all cases.
