About Curves on Surfaces
You can create a curve on surface (COS) in the following ways:
Specify points through which the curve passes on a surface
Drop or project a curve onto a surface, datum plane, or body
Intersect one or more surfaces, bodies, or quilts with one or more surfaces, intent surfaces, datum planes, bodies, or quilts.
However, all points of a COS are constrained to lie on a single surface so that the curve is also on the surface. The curve is a child of the surface.
When you drop a curve on a composite surface, individual COS curves are created for each component of the composite surface. Similarly, you can create a COS by specifying points on the individual components of a composite surface.