An isoperimetric inequality with applications to curve
              shortening

Gage, Michael E.

doi:10.1215/s0012-7094-83-05052-4

Cited by 181 publications

(85 citation statements)

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“…Furthermore, (1.5) implies that d dt (L 2 − 4π A) ≤ 0, which suggests that finite-time extinction is possible (as the isoperimetric inequality L 2 ≥ 4π A forces the isoperimetric difference in brackets to be non-negative and we must have both limits L → 0 and A → 0 at extinction). A further straightforward result for simple closed convex curves is found by using Gage's inequality [2] γ k 2 ds ≥ π L A .…”

Section: T)mentioning

confidence: 99%

“…The tangent velocity is specified by the choice of parameter u and has no effect on the evolution of γ . For this flow, it is known that embeddedness is preserved and that any initially simple closed curve will become convex in finite time [1], then shrink to a point, becoming asymptotically circular as it does so [2][3][4] (for a summary of this problem, see [5]). We refer to the shrinking of a curve to a point at t = T < ∞ as finite-time extinction.…”

Section: Introductionmentioning

confidence: 99%

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A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Dallaston

McCue

2016

Proc. R. Soc. A.

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Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.

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Section: T)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Dallaston

McCue

2016

Proc. R. Soc. A.

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“…In some cases there is partial overlap; these cases we list below: a) axis ratios (for main approximate axes a>b>c we have p=c/a, q=b/a) have been broadly used in the geological literature [10][11][12], however, few rigorous mathematical results are available [13]. b) the isoperimetric ratio I is related to the classical roundness measure of pebbles [14], it is increasingly used in recent works [5,6,15] and there are also mathematical results available [16,17]. c) the number N of mechanical equilibria (i.e.…”

Section: Introductionmentioning

confidence: 99%

Identification of Primary Shape Descriptors on 3D Scanned Particles

Ludmány

Domokos

2018

Period. Polytech. Elec. Eng. Comp. Sci.

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The number of global mechanical equilibria as a shape descriptor (among others, for sedimentary particles) is at the forefront of current geophysical research. Although the technology is already available to provide scanned, 3D images of the particles (appearing as fine spatial discretization of smooth surfaces), nevertheless, the automated identification and measurement of global equilibria on such 3D images has not been solved so far. The main difficulty lies in the algorithmic distinction between local equilibria (associated with the small un-evenness of the pebble’s surface) and global equilibria, associated with the overall shape. The former are easily measured, however, only the latter provide meaningful physical information. Here we provide and illustrate an algorithm to detect global equilibrium points on a finely discretized, polyhedral surface provided by 3D scan of sedimentary particles.

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“…Historically, the curve shortening flow was first proposed in 1956 by Mullins to model the motion of idealized grain boundaries [20]. The study of this flow developed during the 1980s, through the work by Gage (see [7,[9][10][11][12][13][14]) and Hamilton (see [8,17]) on convex plane curves and Grayson (see [15,16] on embedded plane curves, at the time when geometers started using geometric flows to study topological problems.…”

Section: Introductionmentioning

confidence: 99%