269 Citations
Computing critical point evolution under planar curvature flows
- Physics
- 2020
We present a numerical method for computing the evolution of a planar, star-shaped curve under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flows. This…
Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow
- Mathematics
- 2020
Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve $X_0$ into…
Gage’s original normalized CSF can also yield the Grayson theorem
- Mathematics
- 2016
Mimicking Andrews-Bryan’s argument, it is proved in this note that Gage’s original normalized curve shortening flow can also yield the Grayson theorem.
A note on Grayson's theorem
- Mathematics
- 2014
In this note we show a variational proof of Matthew Grayson’s convexification theorem for simple closed curves moving by curvature in the plane. CONTENTS
Noncollapsing in mean-convex mean curvature flow
- Mathematics
- 2012
We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial…
Non-collapsing in mean-convex mean curvature flow
- Mathematics
- 2011
We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial…
The blow up analysis of the general curve shortening flow
- Mathematics
- 2009
It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they…
POLYGON SHORTENING MAKES (MOST) QUADRILATERALS CIRCULAR
- Mathematics
- 2002
We show that an analog of the Gage-Grayson-Hamilton Theorem for curves moving according to their mean curvature holds for the motion of quadrilaterals according to their Menger curva- ture.
Medial Axes and Mean Curvature Motion I: Regular Points
- BiologyJ. Vis. Commun. Image Represent.
- 2002
A set of conditions on the local validity of a medial axis transform and a differential equation for the change of smooth parts of the medial axis when its generating curve evolves under MCM are presented.
References
SHOWING 1-3 OF 3 REFERENCES
BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES
- Mathematics
- 1979
Because of Property 1, any Bonnesen inequality implies the isoperimetric inequality (1). From Property 2, it follows that equality can hold in (1) only when C is a circle. The effect of Property 3 is…
Convex Sets and Their Applications
- Mathematics
- 1982
Fundamentals. Hyperplanes. Helly-Type Theorems. Kirchberger-Type Theorems. Special Topics in E2. Families of Convex Sets. Characterizations of Convex Sets. Polytopes. Duality. Optimization. Convex…