About curves-on-surface
Curves-on-surface are special curves that are drawn in the UV space of a surface, rather than in the XYZ space of the scene. They are usually used to define the line along which to trim surfaces or to form the edge of new surfaces.
Curves-on-surface do not have CVs. They are controlled by moving on-curve edit points.
You can create curves-on-surface by drawing directly on the surface, by projecting existing curves onto a surface, and by intersecting existing geometry with a surface.
Another type of curve-on-surface is a visual curve, produced by tools such as Cross Section Editor, Parting Line, Highlights, and so on. These curves are for display purposes only, and have no geometry attached to them.
Left: Geometry curves-on-surface. Right: Visual curves-on-surface.