The generating function of a 3D CS is a function of the 3 coordinate components of a coordinate 3-tuple. If one of the coordinate components is held fixed to a constant value, the generating function thus restricted to two variables can be viewed as a surface CS generating function (with a surface CS range). If two of the three coordinate components are held fixed, the generating function restricted to one variable can be viewed as a curve CS generating function (with curve CS range). These observations motivate the definitions of coordinate surfaces and curves.
If F is the generating function of a 3D CS, and u = (u0, v0, w0) is in the interior of the CS domain D, then three surface CS generating functions are defined by:
,
, and
.