Straight through the curve

Why does the straight stick need a curved opening?

Swing the stick through the opening in the vertical plane.

The straight stick passes cleanly through the curved opening in the plane.

The straight stick is attached to a vertical axle. The plane is also vertical, but the stick is angled. The ends of the stick are furthest from the axle and the centre of the stick is closest to it. Each of the points on the stick passes through the opening at its own relative distance from the axle. For this reason, the opening in the plane must be curved.

The shape of the opening is a hyperbola. The surface which the straight stick describes as it spins around its axle is a hyperboloid.

It is often difficult to perceive the path that a moving object will take in reality. When moving from one residence to another, you might need several attempts to get the sofas and tables in and out through narrow doorways – and not always do they fit.