A catenary is the shape taken by a hanging chain (the word’s root is “chain”) and has a mathematical definition. Gravity accounts for the shape, kind of a squashed parabola. Inverted as an arch, it is a sturdy self-supporting structure like the St. Louis Gateway Arch.
One would use the catenary to build a smooth road for square wheels. Read this Science News article or check out the Square Wheels Exploratorium book to make your own. I’m building a wood-block arch from some scrap cypress, following the Exploratorium instructions, & will post pictures when I get there. The blanks are all set and just need some precision cutting on my vague table saw.
From the Exploratorium (arch behind illustrative heavy and light chains):