## 265 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…