Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe length-constrained ideal curve flow
137
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A recent article by the first two authors together with B Andrews and V-M Wheeler considered the so-called `ideal curve flow, a sixth order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the $L^2$ sense. Critical in the analysis there was a length bound on the evolving curves. It is natural to suspect therefore that the length-constrained ideal curve flow should permit a more straightforward analysis, at least in the case of small initial `energy. In this article we show this is indeed the case, with suitable initial data providing a flow that exists for all time and converges smoothly and exponentially to a multiply-covered round circle of the same length and winding number as the initial curve.
rate research
Read More
We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles.
We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $Psubset mathbb R^2$ of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and $P$ is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between ACSF and HCS generalizes the link between ACSF and convex-layer decomposition (Eppstein et al., 2017; Calder and Smart, 2020), which is restricted to convex curves. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.
In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.
In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call grid peeling) is the convex-layer decomposition of subsets $Gsubset mathbb Z^2$ of the integer grid, previously studied for the particular case $G={1,ldots,m}^2$ by Har-Peled and Lidicky (2013). The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where $G=mathbb N^2$ is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times $t>0$. We prove that, in the grid peeling of $mathbb N^2$, (1) the number of grid points removed up to iteration $n$ is $Theta(n^{3/2}log n)$; and (2) the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor.
We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $mathbb{R}^ntimes mathbb{R}$, $nge 2$, of the form $(r,y(r))$ or $(r(y),y)$ where $r=|x|$, $xinmathbb{R}^n$, is the radially symmetric coordinate and $yin mathbb{R}$. More precisely for any $lambda>frac{1}{n-1}$ and $mu>0$, we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $frac{r(y)}{1+r(y)^2}=frac{n-1}{r(y)}-frac{1+r(y)^2}{lambda(r(y)-yr(y))}$ in $mathbb{R}$ which satisfies $r(0)=mu$, $r(0)=0$ and $r(y)>yr(y)>0$ for any $yinmathbb{R}$. We will prove that $lim_{ytoinfty}r(y)=infty$ and $a_1:=lim_{ytoinfty}r(y)$ exists with $0le a_1<infty$. We will also give a new proof of the existence of a constant $y_1>0$ such that $r(y_1)=0$, $r(y)>0$ for any $0<y<y_1$ and $r(y)<0$ for any $y>y_1$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
