A CV curve on surface is similar to a plain CV curve, but it lies on a surface. You create it by drawing rather than projecting from a different curve. You can use this curve type for trimming the surface on which it lies.
Trimming a surface with a CV curve on surface
There are two methods for drawing and editing curves on surfaces: drawing in a viewport, or using the Edit Curve on Surface dialog. The choice is useful because you draw in two dimensions, with a mouse or other pointing device, while the curve on a surface can exist in three dimensions. The more complex the 3D surface, the more effort it can require to create and edit a curve on a surface.
Visual feedback can help you draw the curve. The point whose surface you first click is shown as a blue square, and the surface's minimum UV point is shown as a plus sign (+). As you draw the curve, it is displayed interactively in viewports.
When you click to position a CV, the click is projected in the viewport's Z dimension. That is, your click is projected "through the screen" and onto the surface. This is a straightforward way to create a curve on a surface if the portion of the surface where the curve will lie is clearly visible in the viewport. However, this method doesn't let you place CVs on surface locations that are not visible in the viewport (they are on back faces, lie behind other geometry, and so on).
Using the Edit Curve on Surface Dialog
The Edit Curve on Surface dialog lets you edit curves on surfaces as you edit regular curves in a viewport. The main part of the dialog is a two-dimensional view of the surface. The controls provide typical curve editing functions. While you are creating a CV curve on surface, the 2D View toggle controls display of the Edit Curve on Surface dialog.
You can edit the CVs in CV curves on surfaces at the Curve CV sub-object level, as you edit other kinds of curve CVs. You can transform CVs in CV curves on surfaces, but you can't move the CVs off the surface. Using the Curve CV sub-object level is an alternative to editing these CVs by using the Edit Curve on Surface dialog.
To create a CV curve on surface:
The NURBS object must contain at least one surface.
(Create CV Curve On Surface).
Automatic Reparameterization group
The radio buttons in this group box let you choose automatic reparameterization. With reparameterization, the curve maintains its parameterization as you edit it. Without reparameterization, the curve's parameterization doesn't change as you edit it, and can become irregular.
Chooses the chord-length algorithm for reparameterization.
Chord-length reparameterization spaces knots (in parameter space) based on the square root of the length of each curve segment.
When on, displays the Edit Curve on Surface dialog, which lets you create the curve in a two-dimensional (UV) representation of the surface.
Automatic Reparameterization group
The radio buttons in this group box let you choose automatic reparameterization. With reparameterization, the curve maintains its parameterization as you edit it. Without reparameterization, the curve's parameterization doesn't change as you edit it, and can become irregular.
Chooses the chord-length algorithm for reparameterization.
Chord-length reparameterization spaces knots (in parameter space) based on the square root of the length of each curve segment.
Click to display the Edit Curve on Surface dialog, which lets you edit the curve in a two-dimensional (UV) representation of the surface.
To edit multiple curves on a surface, select more than one CV curve sub-object on the same surface, then click Edit.
Displays the Rebuild CV Curve dialog to let you rebuild the CV curve on surface.
Displays the Reparameterize dialog to let you reparameterize the CV curve on surface.
Modify panel 
