Euler's elasticae interface in Mathematica

The elastica can be understood from a number of different aspects, including as a mechanical equilibrium, a problem of the calculus of variations, and the solution to elliptic integrals. In addition, it has a number of analogies with physical systems, including a sheet holding a volume of water, the surface of a capillary, and the motion of a simple pendulum. It is also the mathematical model of the mechanical spline, used for shipbuilding and similar applications, and directly inspired the modern theory of mathematical splines. More recently, the major focus has been on efficient numerical techniques for computing the elastica and fitting it to spline problems. A beautiful family of curves based on beautiful mathematics has constructed by elliptic functions and interfaced by Wolfram Mathematica program.