  Principal curves are defined as parametric curves passing through the
"middle" of a probability distribution in R^d. In addition to the original
definition based on self-consistency, several points of view have been
considered among which a least square type constrained minimization problem.In
this paper, we are interested in theoretical properties satisfied by a
constrained principal curve associated to a probability distribution with
second-order moment. We study open and closed principal curves f:[0,1]-->R^d
with length at most L and show in particular that they have finite curvature
whenever the probability distribution is not supported on the range of a curve
with length L.We derive from the order 1 condition, expressing that a curve is
a critical point for the criterion, an equation involving the curve, its
curvature, as well as a random variable playing the role of the curve
parameter. This equation allows to show that a constrained principal curve in
dimension 2 has no multiple point.
