It occurred to me in my Calc II class today that we only ever revolve functions over a straight line.
One theory I came up with with a friend, that you couldn't do it on a Cartesian plane, and that if the non-straight function would change the scale of the axes, then it could it could be done.
I was wondering if it was possible to do the reverse, to revolve (for simplicity's sake) a line over a curve. (Q1a) A curve over a curve? (Q1b)
If possible, as far as coursework is concerned, when would that material be taught? (Q2)
Thanks!
If I understand you correctly, then, yes, your idea makes sense.
I assume you want to rotate some curve $G$ (the "generator") around another given curve $A$ (the "axis" curve).
Let's start with a point $P$ on $G$. Then, find the point $Q$ on $A$ that is the "foot" of the normal line from $P$ to the curve $A$. The line $PQ$ will be perpendicular to the tangent line of $A$ at $Q$. Form a circle by rotating $P$ around this tangent line. The family of circles contructed this way will form a surface. It's a generalized surface of revolution, with a curved "axis of rotation".
Of course, this construction would go wrong if $A$ were highly curved or if $A$ and $G$ were a long way apart, but it would work OK in "nice" cases.
Here's a picture. The red dashed curve is the axis:
You asked when this might be taught. It's a fairly unusual construction (in my experience, anyway) and it probably would not be taught in any course. Maybe in differential geometry, if you're lucky.