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 may 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 may be viewed as a curve CS generating function (with curve CS range). These observations motivate the definitions of coordinate surfaces and curves. The coordinate surface and coordinate curve concepts are required to specify induced CS relationships and the definition of special coordinate 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
.