Geometric computation of curvature driven plane curve evolutions (Q2719227)

Let \(C\) be a closed smooth plane curve which satisfies the differential equation NEWLINE\[NEWLINE\frac{\partial C}{\partial t}(s,t)=F(\kappa(s,t))\mathbf{N}(s,t),\tag{1}NEWLINE\]NEWLINE where \(\mathbf{N}(s,t)\) is the inner normal vector to the curve at \((s,t)\), \(\kappa\) is the curvature and \(F\) is a nondecreasing function with \(F(0)=0\). In addition it is assumed that either \(x\to F(x^3)\) is Lipschitz with Lipschitz constant \(\leq 1\) or \(F(x)=x^\gamma\) for \(\gamma \geq 1/3\). NEWLINENEWLINENEWLINEThe paper presents a numerical algorithm to solve (1) by following a geometrical approach which avoids some drawbacks of finite difference schemes. A generalization of a theoretical algorithm for moving hypersurface by a power of Gauss curvature, previously studied by \textit{H. Ishii} [GAKUTO Int. Ser., Math. Sci. Appl. 14, 198-206 (2000; Zbl 0987.53027)], is generalized and implemented in the plane for nonconvex curves and for more general functions of the curvature. NEWLINENEWLINENEWLINEThe evolution of circles and some other closed convex curves are displayed in the last section.