Euclidean Shortening Flow

In Euclidean space, the equations that give rise to the geometric heat equations are the curvature deformation flow for a 2D curve, $\mathcal{C}$, and its 3D extension, the mean curvature deformation flow for a surface, $\mathcal{S}$,

$$\frac{\partial \mathcal{C}}{\partial t} = \kappa \overrighta... ...qquad \frac{\partial S}{\partial t} = \kappa \overrightarrow N$$

In the 3D case, $\kappa$ is the mean curvature of the surface defined in terms of principal curvatures $\kappa _1$ and $\kappa _2$ [13],

$$\kappa = \frac{\kappa _1 + \kappa _2}{2}$$

It is well-known in the literature that the 2D evolution is the Euclidean curve shortening flow in the sense that perimeter shrinks as fast as possible. In fact, it is possible to obtain this flow from first principle by minimizing the proper length functional [15]. But what is the link with the heat equation? The heat equation is

$$\psi _t = \psi _{rr} + \psi _{r'r'}$$

for any $r$ and $r'$ perpendicular. In our case, the $r'$ term is in direction of the gradient to the curve and it vanishes if the signed distance function [1] is used as higher dimensional embedding function. Also, using level set formulation [1],[5], the curve flow can be rewritten as

$$\psi _t = \psi _{rr}$$

where r is perpendicular to $\nabla \psi$, i.e., r is along the curve. This is derived in [1]. Hence, the flow does not smooth in all direction. It does not smooth across structure but along. Therefore, the curvature deformation flow is anisotropic and a geometric case of the heat equation.