Steepest Descent as a Tool to Find Critical Points of ∫ k2 Defined on Curves in the Plane with Arbitrary Types of Boundary Conditions

We will explicitly compute the gradient of the total squared curvature functional on a space of parametrized curves of fixed or variable length satisfying arbitrary types of boundary conditions. We show how to turn the space of such curves into an infinite dimensional submanifold of an inner product space. The steepest descent will then be along the integral curves of the negative gradient vector field in this manifold. We will derive the gradients and the corresponding flow equations. In conclusion we use computer graphics to illustrate this process by following one such trajectory starting close to an unstable critical point and ending at a stable critical point.