A Curve-on-Surface is different from a normal NURBS or Bezier curve used for modeling and is a fundamental part of the trimming workflow in Alias. A regular curve is defined in the x,y,z modeling space, and is free to move in any direction. A CoS is defined in the UV parameter space of the surface it lies on, and cannot move away from that surface. A CoS also has only Edit Points, no CVs.
In Alias, the only way to trim (face) a surface is to have a Curve-on-Surface (CoS) defining the boundaries of the trim regions. Curves-on-Surface are also important in surfacing operations on the edges of trimmed surfaces. They provide better anchoring of CVs and blend points when attaching to the edges of trimmed surfaces.
