The Isoperimetric Quotient of a Convex Body Decreases Monotonically Under the Eikonal Abrasion Model

Abstract

We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically.2010 Mathematics Subject Classification. 35Q85, 35Q86, 52A39.

Tracking Critical Points on Evolving Curves and Surfaces

“…Unlike curvature-driven flows, (19) does not preserve the smoothness of the evolving manifold. As long as r(t) remains smooth (approximately until t = 1700 on Figure 7), N (t) remains constant and once r(t) becomes non-smooth, N (t) decreases monotonically [15]. However, the coupling between N ∆ (t) and N (t) is in this case rather different: here the downward jumps of N (t) are not coupled to resonance-like (A-type) events in the evolution of N ∆ (t), rather, both evolutions have a downward trend (see our example illustrated in Figure 7).…”

“…The time dependence N (t) of the number of global critical points has been broadly investigated in various evolution equations [10,12,14,15,22,25]. Our goal here is rather different: instead of studying any particular evolution equation (which we will use only as illustrations) we focus on some universal features relating N (t) to N ∆ (t).…”

A new workflow for the automated measurement of shape descriptors of rocks

“…Traditional measurement techniques often incorporate personal factors or rely on the verbal characterization of the shape (Wentworth, 1923;Boggs, 2001) to approximate these values. The number of stable (S) and unstable (U ) mechanical equilibria of a particle as a shape descriptor gained significant attention recently (Domokos et al, 2009; Miller et al, 2014;Domokos et al, 2015;Novák-Szabó et al, 2018) and it is not only insensitive for small measurement errors, but it also has a rich mathematical literature (Grayson, 1987;Domokos et al, 2015;Domokos and Lángi, 2019).…”

“…Traditional measurement techniques often incorporate personal factors or rely on the verbal characterization of the shape (Wentworth, 1923;Boggs, 2001) to approximate these values. The number of stable (S) and unstable (U ) mechanical equilibria of a particle as a shape descriptor gained significant attention recently (Domokos et al, 2009; Miller et al, 2014;Domokos et al, 2015;Novák-Szabó et al, 2018) and it is not only insensitive for small measurement errors, but it also has a rich mathematical literature (Grayson, 1987;Domokos et al, 2015;Domokos and Lángi, 2019).It is beneficial to switch from the traditional hand-measurements to automated shape analysis of the particles to avoid personal bias. Recently, several works appeared aiming to reduce subjectivity by automatically calculating shape properties from 2D digital images of the particles (Roussillon et al, 2009;Durian et al, 2007;Cassel et al, 2018;Cheng et al, 2018), 25 3D laser scanning (Hayakawa and Oguchi, 2005; Anochie-Boateng et al, 2013) or X-ray CT (Deiros Quintanilla et al, 2019).…”

Identification of Primary Shape Descriptors on 3D Scanned Particles

“…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.…”