Rotational Crofton formulae for Minkowski tensors and some affine counterparts

Svane, Anne Marie; Jensen, Eva B. Vedel

doi:10.1016/j.aam.2017.05.009

Cited by 5 publications

(5 citation statements)

References 23 publications

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“…The aim of the present work is to prove a complete set of Crofton formulae for the (generalized) tensorial curvature measures. This complements the particular results for (extrinsic) tensorial curvature measures and Minkowski tensors obtained in [14] and [29]. The current approach is basically an application of the kinematic formulae for (generalized) tensorial curvature measures derived in [15].…”

Section: Introductionsupporting

confidence: 69%

Crofton Formulae for Tensorial Curvature Measures: The General Case

Hug

Weis

2017

Analytic Aspects of Convexity

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The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. In a previous work, we obtained kinematic formulae for all (generalized) tensorial curvature measures. As a consequence of these results, we now derive a complete system of Crofton formulae for such (generalized) tensorial curvature measures. These formulae express the integral mean of the (generalized) tensorial curvature measures of the intersection of a given convex body (resp. polytope, or finite unions thereof) with a uniform affine k-flat in terms of linear combinations of (generalized) tensorial curvature measures of the given convex body (resp. polytope, or finite unions thereof). The considered generalized tensorial curvature measures generalize those studied formerly in the context of Crofton-type formulae, and the coefficients involved in these results are substantially less technical and structurally more transparent than in previous works. Finally, we prove that essentially all generalized tensorial curvature measures on convex polytopes are linearly independent. In particular, this implies that the Crofton formulae which we prove in this contribution cannot be simplified further.

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Section: Introductionsupporting

confidence: 69%

Crofton Formulae for Tensorial Curvature Measures: The General Case

Hug

Weis

2017

Analytic Aspects of Convexity

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“…6.4], recent progress for scalar-and measure-valued valuations and further references are provided in [82,83,39], applications to stochastic geometry are given in [37,35,34], where translative integral formulae for tensor-valued measures are established and applied. Rotational Crofton formulae for tensor valuations have recently been developed further by Auneau et al [6,7] and Svane & Vedel Jensen [77] (see also the literature cited there), applications to stereological estimation and bio-imaging are treated and discussed in [58,87,78]. Various other groups of isometries, also in Riemannian isotropic spaces, have been studied in recent years.…”

Section: Introductionmentioning

confidence: 99%

“…Integral geometric Crofton formulae for general Minkowski tensors have been obtained in [44]. A specific case has been further studied and applied to problems in stereology in [54], for various extensions see [46] and [77]. A quite general study of various kinds of integral geometric formulae for translation invariant tensor valuations is carried out in [15], where also corresponding algebraic structures are explicitly determined.…”

Section: Introductionmentioning

confidence: 99%

Kinematic formulae for tensorial curvature measures

Hug

Weis

2018

Annali di Matematica

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Tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. We prove a complete set of kinematic formulae for such tensorial curvature measures on convex bodies and for their (nonsmooth) generalizations on convex polytopes. These formulae express the integral mean of the tensorial curvature measure of the intersection of two given convex bodies (resp. polytopes), one of which is uniformly moved by a proper rigid motion, in terms of linear combinations of tensorial curvature measures of the given convex bodies (resp. polytopes). We prove these results in a more direct way than in the classical proof of the principal kinematic formula for curvature measures, which uses the connection to Crofton formulae to determine the involved constants explicitly.Date: December 26, 2016. 2010 Mathematics Subject Classification. Primary: 52A20, 53C65; secondary: 52A22, 52A38, 28A75.

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“…The Crofton formula has attracted much attention in calculus geometry and has been applied in the study of hyperbolic space surfaces [1,2] and Minkovski tensors [3] . In the Euclidean plane, the length of the curve can be calculated using the Crofton formula.…”

Section: Introductionmentioning

confidence: 99%

Calculation and Implementation of Crofton Formula

Zhang¹,

Chu²,

Li³

2019

dtetr

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This paper introduces the Crofton formula of differential geometry, and clarifies the meaning of each variable. Furthermore, in order to verify the identity of the Crofton formula, the lengths of several special curves have been accurately calculated by Matlab programming, and have been compared with the actual lengths. At the same time, the Crofton formula is transformed from the calculation of the double integral to the calculation of the micro-element, so that it is easier to find the approximate solution to the general curve by Matlab programming. It is concluded that this method can be used to find an approximate solution for the length of any general curve. This template explains and demonstrates how to prepare your camera-ready manuscript for publisher. The best is to read these instructions and follow the outline of this text. Please make the page settings of your word processor to A4 format (21 x 29, 7 cm or 8 x 11 inches); with the margins: bottom 1.5 cm (0.59 in) and top 2.5 cm (0.98 in), right/left margins must be 2 cm (0.78 in). Your manuscript will be reduced by approximately 20% by the publisher. Please keep this in mind when designing your figures and tables etc.

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Rotational Crofton formulae for Minkowski tensors and some affine counterparts

Cited by 5 publications

References 23 publications

Crofton Formulae for Tensorial Curvature Measures: The General Case

Crofton Formulae for Tensorial Curvature Measures: The General Case

Kinematic formulae for tensorial curvature measures

Calculation and Implementation of Crofton Formula

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