Is there a generalized method of rotation for curves? I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution.  But has anyone come up with a general method to take a real valued function in $R^2$ and rotate about another function in $R^2$ that is not necessarily linear?  I assume the generalized definition would take a point off the curve to the point on the other side of the curve the same distance off the curve along a line perpendicular to the curve at some point.  The alternative definition I thought of was given in an answer below, but that is not what I'm looking for.  I want to be able to rotate a curve about another curve geometrically with or without an established coordinate plane, which is why I assumed the definition above.
Trying to make this more precise:
My definition takes a curve $C$ and finds the slope of the normal line at point $(x_o,y_o)$.  Supposing the slope found is $m$, the normal line is $y=m(x-x_o)+y_o$.  Find the point(s) $(x_o,y_o)$ for which this line intersects the point/curve to be rotated.  Find the distance along this line between $C$ and the point to be rotated.  Then, traversing the line in the opposite direction from $C$, find the point that same distance away from $C$.  This yields the rotated point.
Using the definition I give and use Curve $C$ as an axis:
 1.  Is the relation between a point in $R^2$ and its image after rotation a function?
 2.  Does this depend on whether $C$ represents a real valued function?  For example, rotation about a circle is not a function while rotation about a parabola is?
 3.  Does this yield a well-defined surface of revolution?
 4.  Could such a rotation yield interesting results, e.g., transforming a smiley face into a sad face, or turning a one kind of conic into another?
 5.  Supposing this definition cannot yield a well-defined surface of revolution, as some have suggested, what definition could?
 6.  Are there helpful links or articles that address any of these issues?
A counterexample to the conjecture of(1):  Take $C$ to be the unit circle centered at the origin and rotate $(0,2)$ about $C$.

Depending on the purpose you have in mind, rotation of the function $f$ about the function $g$ may be equivalent to the rotation of $f-g$ about the $x$-axis. 
In particular, this works for finding the volume of the solid defined by the rotation of $f$ about $g$.
