Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimes also known as rollers.
The "width" of a closed convex curve is defined as the distance between parallel lines bounding it ("supporting lines"). Every curve of constant
width is convex. Curves of constant width have the same "width" regardless
of their orientation between the parallel lines. In fact, they also share the same
perimeter (Barbier's
theorem). Examples include the circle (with largest
area), and Reuleaux triangle
(with smallest area) but there are an infinite number. A
curve of constant width can be used in a special drill chuck to cut square "holes."
A generalization gives solids of constant width. These do not have the same surface area for a given width, but their shadows are curves of
constant width with the same width!
