While combing around my notes looking for other possible examples for this question, I chanced upon another one of my unsolved problems:

Cycloidal curves are curves generated by a circle rolling upon a plane or space curve. It's not too hard to derive the required parametric equations if the fixed curve is a plane curve, but I've had some trouble deriving the corresponding expression for space curves.

More specifically, here is the particular problem I was concerned with: consider a (cylindrical) helix:

$$\begin{align*}x&amp;=a\cos\;t\\y&amp;=a\sin\;t\\z&amp;=ct\end{align*}$$

and imagine a circle of radius $r$ whose plane is always perpendicular to the x-y plane rolling on the helix, starting at the point $(a,0,0)$ ($t=0$). Imagine a point in the plane of the circle at a distance $hr$ from the center. What are the parametric equations for the locus of the point?

The two obvious pieces of information I have are that the center of the circle also traces a helix, whose parametric equation differs from the original by a vertical shift of $r$ (per Tony, that was an erroneous assumption), and that the expression for the arclength of the helix, $s=\sqrt{a^2+c^2}t$, should figure into the final parametric equations. Otherwise, I'm not sure how to start.

How does one derive parametric equations for the "helical cycloid"?



The physical model I had in mind was a screw ramp winding around a cylinder. Supposing that there was a car that needed to go to the top of the cylinder by driving on the ramp, and supposing that a spot is placed on one of the car's wheels, what are the equations for the locus of the spot?


Question: What's the first physics textbook for undergraduate self-learner？
Answer: You can study "Modern physics" by Arthur Beiser. It is a very good book for undergraduate students. And also you can study the Feynmann Lecture series.