Evolution of curves on a surface driven by the geodesic curvature and external force

Abstract

We study a flow of closed curves on a given graph surface driven by the geodesic curvature and external force. Using vertical projection of surface curves to the plane we show how the geodesic curvature-driven flow can be reduced to a solution of a fully nonlinear system of parabolic differential equations. We show that the flow of surface curves is gradient-like, i.e. there exists a Lyapunov functional nonincreasing along trajectories. Special attention is placed on the analysis of closed stationary surface curves. We present sufficient conditions for their dynamic stability. Several computational examples of evolution of surface curves driven by the geodesic curvature and external force on various surfaces are presented in this article. We also discuss a link between the geodesic flow and the edge detection problem arising from the image segmentation theory.

Invariant submanifold flows

“…For any of the Euclidean invariant normal plane curve flows C t = J n listed in Example 4.3, we have, according to (3.33), thereby recovering formulas used in Gage and Hamilton's analysis, [20]; see also Mikula andŠevčovič, [39,40,41]. Finally, for the mKdV flow, J = κ s ,…”

“…While a number of particular examples have been worked out by direct computation, e.g., in [20,39,40,41], many cases of interest have yet to appear in the literature, owing in part to the complexity of the required calculations. Therefore, it is worth developing general, practical computational tools to facilitate this often tedious task.…”

Evolution of plane curves with a curvature adjusted tangential velocity

“…This result can be considered as an improvement of that of [14] in which satisfactory results were obtained only for the case when β = β(k) is linear or sublinear function with respect to k. Next, in the series of papers [16][17][18], Mikula and the first author proposed a method of asymptotically uniform redistribution. In terms of our notation, they derived (3.3) with ϕ ≡ 1 and nontrivial relaxation function ω(t) for a general class of normal velocities of the form β = β(x, ν, k).…”

“…Under the assumption ω ≡ 0 with suitable κ 1 , κ 2 , redistribution of grid points becomes asymptotically uniform [16][17][18].…”

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

“…The case when b = − 1 2 k 3 arises from the model of the Euler-Bernoulli elastic rod -an important problem in structural mechanics [6,12,11]. The evolutionary models having the normal velocity of the form (1.1) are often adopted in image segmentation where elastic and geodesic curves are used in order to find image objects in an automatic way [19,7,20,27,29]. By our method we are able to handle a regularized backward mean curvature flow in which b is a decreasing function of the curvature k like e.g.…”