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Suppose one intersects unit-radius solid cylinders in $\mathbb{R}^3$, with each cylinder axis passing through the origin. For example, two such cylinders produce the Steinmetz solid.

But if we imagine starting with one cylinder with axis aligned along the $x$-axis, and spin that axis about $z$, obtaining an infinite number of cylinders all of whose axes lie in the $xy$-plane, the resulting intersection is a unit-radius ball:


          Cylinders3
If we set a unit direction vector $u$ along each cylinder axis, the spinning walks the $u$ vector tips around a great circle on the sphere $S$ of directions.

Now imagine an arbitrary closed curve $C$ on $S$. Form the intersection $I(C)$ of the cylinders determined by all the direction vectors lying on $C$. Again all cylinders have unit radius, and all axes are through the origin. My question is:

Q. For which $C$ will $I(C)$ be the unit-radius ball?

This might be straightforward, but I am not seeing it...

An analogous question can be posed in $\mathbb{R}^d$, with each cylinder a line cross a $({d{-}1})$-ball.

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As far as I can see, a necessary and sufficient condition is that $C$ should intersect each great circle. That is, a cylinder with axis $u$ excludes the points just outside the unit ball in the direction of the unit vector $v$ iff $u$ and $v$ are perpendicular. –  Andreas Blass Jun 29 at 21:14
    
@AndreasBlass: Nice, Andreas! And I think that generalizes to higher $d$. If you make that an answer, I'll accept it. Or just leave it as a comment, if you prefer. –  Joseph O'Rourke Jun 29 at 21:48
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In particular, $C$ cannot be strictly contained in any hemisphere $H$, for then it would not intersect the great circle determining $H$. Perhaps that is a more geometric formulation of your condition. –  Joseph O'Rourke Jun 30 at 0:13

1 Answer 1

up vote 6 down vote accepted

At the OP's suggestion, I'm promoting my comment to an answer.

A necessary and sufficient condition is that $C$ should intersect each great circle. That is, a cylinder with axis $u$ excludes the points just outside the unit ball in the direction of the unit vector $v$ iff $u$ and $v$ are perpendicular.

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