Hyperboloid as Ruled Surface

When our daughter-in-law made wheat shocks as center-pieces for hers and our son’s fall-themed wedding reception, I naturally could not help pointing out the age-old observation that they represented a hyperboloid of one sheet. This was naturally greeted with the usual groans, but the thought stayed with me as I realized I had never proved this mathematically to myself. And so I did.

See the Hyperboloid as Ruled Surface.

(Updates 10/9/2020, 9/19/2022) Spinning Rod Demo, Spinning Umbrella

(Update 10/9/2020) Spinning Rod Demo

It dawned on me after watching the following the surprising video, that the demonstration was actually showing the generation of a hyperboloid ruled surface and the vertical plastic sheet was just a hyperbolic cross-section of the surface.  It looks like his mathematical derivation is similar to mine.

From Futility Closet: Throwing a Curve, 9 October 2020:

“In 2009, mathematician Jeff Chyatte and his colleagues at Maryland’s Montgomery College built a mathematical sculpture: An inclined rod is connected at its center to a horizontal arm, which is connected to a rotating vertical axis. As the axis rotates, the rod passes through a vertical plane.

What shape does the rod cut in the plane? Perhaps surprisingly, it’s a hyperbola. See the video above for an explanation. Chyatte’s sculpture was displayed at Washington’s Touchstone Gallery with the title “Theorem.”

(“Just Passing Through,” Math Horizons 16:4 [April 2009], 16.)”

(Update 9/19/2022) Spinning Umbrella

Here is another, more dramatic example.