Suspension bridges, spiderwebs, the lines between ship and pier. The catenary's signal feature is that the force involved in holding it up is aligned with a tangent to the curve, at all points. What this means is that there's no sideward force, as there would be with a semicircular, Roman arch. I knew none of this when I made my first arch. The legs lean inward, increasing their lean, but the increasing sideward load is countered by an equal lean in the other leg.
Gravity is a sand sculptor's friend, holding the sand together.
This is a basic shape for any sand sculpture that flies. Being a compression structure, the catenary arch requires sufficient cross-section to bear the imposed load. This is true along the arch's entire length, and I've had them collapse due to soft spots in the legs. It's a balancing act. The legs must be gradually thinned so that no one area overwhelms another.