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: Using a ski helmet for winter biking
Answer: If you're thinking about wearing a ski helmet for biking, for what it's worth the Snell RS-98 recreational skiing and B-95 biking helmet standards are nearly identical, word for word, including things like visibility requirements and anatomical coverage. There are some small differences in the drop test. The energy of the flat anvil drop test in the B-95 standard is 110J vs. 100J for the RS-98 ski standard.  The hemispherical anvil drop test is 80 J for RS-98 and 72 for B-95. Bike helmets are tested against the point of a right angle "kerbstone" anvil, but ski helmets are tested against an "edge anvil" 6.3mm wide -- basically a decapitating machine -- and at higher forces than the kerbstone test too. Both bike and ski helmets have to be tested after being chilled and soaked, but bike helmets have to be tested after being cooked at 50 C (122 F).

The reasonable conclusion is that a Snell certified ski helmet is probably a suitable choice for winter biking, but should not be used after exposure to extreme heat.  Yes, a certified ski helmet might pass the 100J flat anvil test but not the 110 J test, but it probably does.

As for ASTM 2040, who can say? It's not available for free inspection. But it appears that the requirements for recreational ski helmets and bike helmets are pretty close.