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 $C$.