Euler's elastica

Euler's elastica describes the shape of a flexible, elastic rod or curve that minimizes energy by balancing bending and tension forces. Imagine a thin, bendable strip of material: it naturally adopts a smooth curve that minimizes the total bending energy while satisfying boundary constraints. Mathematically, elastica curves arise from calculus of variations, leading to differential equations that govern their shape. These curves are important in fields like engineering, computer vision, and biology, helping us understand phenomena from the bending of fibers to the design of flexible structures.