# Turning 180° on a rail

A vaguely U-turned rail is installed on the ground such that a bar with two ends attached on the rail moves along it to rotate 180°. Like this:

(Sorry for the poor image. I just drew it with MS Paint.)

If the rail can be concave, what would be the shape of the rail if the area inscribed by the rail and the line formed by connecting the lower ends of the bar in the initial and final positions as shown in the image above should be minimized (such that the ending position of the bar is directly on the right of the starting position)?

I thought of a semicircular shape, but I'm not sure if it is minimum. How can it be proved that the perimeter/area must be the minimum?

I just thought of this when I had a bad sleep last night.

• Your polyline shape with shortest lengths leads to 3 units of rail and one square-unit of area. A circle of diameter 1 leads to more rail ($\pi$ units), but less area ($\frac\pi4$). Feb 25, 2017 at 12:52
• It's actually $\pi/2$ units length, isn't it? @HagenvonEitzen Feb 25, 2017 at 13:24
• Can it be proved that either is the shortest/smallest?
– SOFe
Feb 25, 2017 at 13:39