Spatial Stit Tessellations: Distributional Results for I-Segments

Thäle, Christoph; Weiß, V.; Nagel, Werner

doi:10.1239/aap/1346955258

Cited by 5 publications

(7 citation statements)

References 25 publications

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“…We start by determining the marginal distribution Q (j) β,t , that is, the birth time distribution of the typical V jweighted maximal k-polytope of Y t . Our next proposition largely extends and unifies earlier results for the special case k = d − 1 and j = 0 in [37] and d = 3, k = 1 and j ∈ {0, 1} in [42].…”

Section: Markov Properties Of Typical Maximal Polytopes and Their Bir...supporting

confidence: 86%

“…d is strictly increasing for all d ≥ 3. In contrast, for the mean number of internal vertices on the typical length-weighted maximal segment we have that EN In the planar case d = 2, as mentioned above, the probabilities p 1,0 (n) are known from [20,39], whereas for d = 3 the formula for p 1,0 (n) has been established in [42] by different methods. Our approach in the present paper is more general and allows to deduce the corresponding formula also for the length-weighted maximal segment as well as to deal with arbitrary space dimensions.…”

Section: The Number Of Internal Vertices On Maximal Segmentsmentioning

confidence: 99%

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A Mecke-type formula and Markov properties for STIT tessellation processes

Nagel¹,

Nguyen²,

Thaele³

et al. 2017

ALEA

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Section: Markov Properties Of Typical Maximal Polytopes and Their Bir...supporting

confidence: 86%

Section: The Number Of Internal Vertices On Maximal Segmentsmentioning

confidence: 99%

A Mecke-type formula and Markov properties for STIT tessellation processes

Nagel¹,

Nguyen²,

Thaele³

et al. 2017

ALEA

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“…From each such vertex there are precisely 2d emanating edges and each edge has two vertices as its endpoints (see also [18,Equation (16) To compute N 1 (t) we apply Theorem 7.8 to deduce that The previous result can be used, in particular, to determine the expected length of the typical maximal spherical segment of a splitting tessellation Y t on S d . Let us emphasize that in contrast to the expected length of the typical maximal segment in a Euclidean STIT-tessellation (see [33] for the planar case, [56] for the spatial case and [38,53] for general space dimensions), the expected length of the typical maximal spherical segment of a splitting tessellation Y t on S d is universal and does not depend on the underlying direction distribution κ. Especially, if d = 2, this reduces to 2πt t 2 + 1 − e −t .…”

Section: Typical Maximal Spherical Facesmentioning

confidence: 99%

Splitting tessellations in spherical spaces

Hug¹,

Thäle²

2019

Electron. J. Probab.

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The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension d ≥ 2 is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the (d − 1)-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension k ∈ {1, . . . , d − 1} are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation. From now on we call elements of S d−1 great hyperspheres of S d . For a set B ⊂ S d , let us denote by S d−1 [B] := {S ∈ S d−1 : S ∩ int(B) = ∅} the set of all great hyperspheres of S d which have non-empty intersection with the interior int(B) of B (here the interior refers to the topology of S d ). Definition 2.1. A Borel measure κ on S d−1 is called regular if κ({S ∈ S d−1 : e ∈ S}) = 0 for e ∈ S d .Clearly, ν d−1 and any measure which is absolutely continuous with respect to ν d−1 is regular. The assertion of the following lemma will be used repeatedly. For S ∈ S d−1 we denote by S + , S − the two closed half-spheres bounded by S. Since we shall always work with statements symmetric in S + , S − , we do not have to specify this further. Lemma 2.2. If κ is a regular Borel measure on S d−1 and c ∈ P d , then κ({S ∈ S d−1 : S ∩ c = ∅, c ⊂ S + or c ⊂ S − }) = 0 .This remains true for an arbitrary compact set C ⊂ S d in place of c if κ is absolutely continuous with respect to ν d−1 .

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“…Tessellations of that kind arise for example by subsequent cell division. Among these models the iteration stable or STIT tessellations are of particular interest, because of the number of analytically available results, see [7], [5], [1], [12], [11] and the references therein. They may serve as a reference model for crack and fissure structures or for processes of cell division.…”

Section: Introductionmentioning

confidence: 99%

Column Tessellations

2015

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A new class of non facet-to-facet random tessellations in three-dimensional space is introduced -the so-called column tessellations. The spatial construction is based on a stationary planar tessellation; each cell of the spatial tessellation is a prism whose base facet is congruent to a cell of the planar tessellation. Thus intensities and various mean values of the spatial tessellation can be calculated from suitably chosen parameters of the planar tessellation.Keywords: combinatorial topology, random tessellations, stochastic geometry. THE CONTEXT FOR OUR NEW MODELRandom tessellations are classical structures considered by stochastic geometers. Two standard models are the Poisson hyperplane and Poisson Voronoi tessellations (Schneider and Weil, 2008;Chiu et al., 2013). In the planar case these tessellations are side-to-side, that is, each side of a polygonal tessellation cell coincides with a side of a neighbouring cell. In their three dimensional versions they are facetto-facet, meaning that each facet of a polyhedral cell coincides with a facet of a neighbouring cell.In recent years there has been a growing interest in tessellation models that do not have the stated coincidence for all sides or facets. A first systematic study of the effects when a three-dimensional tessellation is not facet-to-facet is given in Weiss and Cowan (2011). A recent study presented in Cowan and Thäle (2014) looks in depth at the planar case, building on results from the 1970s when non side-toside tessellations first attracted attention.Tessellations of that kind arise for example by cell division. Among these models, the iteration stable or STIT tessellation is of particular interest because of the number of analytically available results (Nagel and Weiss, 2005;Mecke et al., 2008;Thäle et al., 2012;Cowan, 2013;Thäle and Weiss, 2013, and the references therein). It serves as a reference model for geological crack and fissure structures (Mosser and Matthäi, 2014) and might have application in the process of biological cell division. The development of new model classes is important for further applications to random structures in materials science, geology and biology -and the current paper contributes to that aim.We consider a new class of non facet-to-facet spatial tessellations, whose construction is based on a stationary planar tessellation Y having convex polygonal cells. From each polygonal cell z of Y we form an infinite column perpendicular to the plane E in which Y lies. Any cross-section of the column parallel to E is congruent to z. To create a spatial tessellation, each infinite column is intersected by many such cross-sections, thereby dividing the column into cells. The spatial cells which arise are prisms and their polygonal base facets (located at the cross-sections) are translations (in the third dimension orthogonal to E ) of the cells of Y . The resulting three-dimensional tessellation Y is called a column tessellation. Column tessellations could be useful to describe crack structures in geology, as for e...

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Spatial Stit Tessellations: Distributional Results for I-Segments

Cited by 5 publications

References 25 publications

A Mecke-type formula and Markov properties for STIT tessellation processes

A Mecke-type formula and Markov properties for STIT tessellation processes

Splitting tessellations in spherical spaces

Column Tessellations

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