This report discusses the problem of recovering 3D motion and structure for rigid curves. For this purpose, we use long monocular sequences of images of the curve and compute some set of derivatives (up to the second order) that are defined on the so-called spatio-temporal surface generated by the curve. For general 3D rigid curves, there is exactly one constraint for each image point that relates these derivatives to the kinematic screw and its first order time derivative. These equations derive directly from the perspective equations. However, this theory ignores (as does the projective model of cameras) the fact that the viewed object must be entirely in front of the focal plane. In this paper, we provide a complete description (up to second order) of these conditions and of their geometrical consequences for 3D reconstruction. Moreover, the relations between the geometry of the image curve and the kinematic screw are investigated. We show that ignoring these constraints leads to erroneous solutions with a high probability even when using perfect data. Then, an algorithm which takes advantage of these constraints is proposed. The accuracy of the obtained results is demonstrated on both synthetic and real image sequences.