ImageAnalStereol2009;28:69-76
OriginalResearchPaper
MOMENTSOFTHELENGTHOFLINESEGMENTSINHOMOGENEOUS
PLANAR STITTESSELLATIONS
CHRISTOPH TH¨ALE
DepartmentofMathematics,UniversityofFribourg, CH-170 0Fribourg,Switzerland
e-mail: christoph.thaele@unifr.ch
(AcceptedFebruary20,2009)
ABSTRACT
Homogeneous planar tessellations stable under iteration ( STIT tessellations) are considered. Using recent
resultsaboutthe jointdistributionofdirectionandlengt hof the typical I-,K- andJ-segmentwe proveclosed
formulasfor the first, second and higher moments of the lengt h of these segments given their direction. This
especiallyleadstothemeanvaluesandvariancesofthesequ antitiesandmeanvaluerelationsaswellasgeneral
momentrelationships. Moreover,therelationbetweenthes e meanvaluesandcertainconditionalmeanvalues
(andalsohighermoments)isdiscussed. Theresultsare also illustratedforseveralexamples.
Keywords:conditionaldistribution,iteration(nesting) ,linearsegments,meanvaluerelation,moments,random
tessellation,stability,stochastic geometry.
INTRODUCTION
Random tessellations are nowadays used for
modeling several structures which arise for example
in material sciences, biology and medical sciences.
The most popular models are Poisson line or plane
tessellations or Poisson-Voronoi tessellations (see
Stoyanetal.,1995).Thesemodelsareontheonehand
side mathematically feasible and useful in practice
on the other. More complex models can be obtained
by applying certain operations on given tessellations.
These are for example superposition or iteration. The
latter operation leads to a relatively new model for
random tessellations, the so-called STIT model. The
name STIT refers to their characteristic property:
they arestable under iteration. This property will be
explainedin somedetailin the nextsection.
Several geometric quantities were calculated for
thismodelbynow.Thisincludesmeanvalueformulas
in 2D and 3D and length distributions of several
linear segments. For the latter Mecke et al.(2007)
calculated also first and second moments, but only
for the isotropic case. They especially observed that
the variance of the length of the so-called typical I-
segmentisinfinite.Thesameobservationwasmadein
Th¨ ale (2008) also for two anisotropic examples. It is
oneaimofthisshortpapertoundertakeadeeperstudy
of this phenomenon. We will show that the variance
(and also other higher moments) of the length of the
typicalI-segmentdoesnotexistfor anyhomogeneous
planarSTITtessellation,whichshows,thattheselinear
segments are in some sense very long. In contrast
to this result we will be able to show that for the
length of the typical K- andJ-segment allmomentsexist. Moreover, we will derive explicit formulas for
all moments of the quantities in question for arbitrary
homogeneousplanarSTIT tessellations.
The other aim of the paper is the study of mean
valuerelations.In particularweareinterestedin mean
values for the lengths of the typical I-,K- andJ-
segment having a fixed direction and their relation to
the mean values without the directional conditioning
(this could be of some interest for stereological
questions). At this point the directional distribution of
the tessellation plays an important role and we will be
able to extend the mean value formulas to the case,
wherethetypical I-,K-orJ-segmentisreplacedbythe
typicalI-,K-orJ-segmentwithafixedgivendirection.
Moreover, the known formulas can be recovered by
averagingoverallpossibledirections.Thisobservation
is a new feature of STIT tessellations, which can only
be observed, when the anisotropic case is studied and
this was notinvestigateduntil the recentworksMecke
(2008) and Th¨ ale (2008). We will furthermore obtain
a general moment relationship for the conditional and
unconditionallength distribution ofthe segments.Our
resultsaredemonstratedonseveralconcreteexamples,
inparticularweconsidertheisotropiccaseandconfirm
again the results of Mecke et al.(2007). Also the new
rectangular case and a case with unequal weights is
discussed.
PLANARSTITTESSELLATIONS
By a planar tessellation we mean a subdivision
of the plane into a locally finite union of convex
polygons, which intersect only in their boundaries.
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The family of such tessellations will be denoted by
T. Such tessellations can also be described as the
union of their cell boundaries. This allows us to
consider a tessellation as a closed subset of R2. We
willfollowthispathhere.Denotingby Ttherestriction
of Matheron’s σ-algebra ( cf.Stoyanet al., 1995) to
T, we call a random variable Φwith values in the
measurablespace [T,T]a randomplanartessellation.
Forx∈R2wedenoteby Txthetranslationof R2bythe
vector −x.Txinducesalsoanoperationonthespaceof
tessellations, also denoted by Tx, byTxΦ=Φ−x. We
say that a random tessellation Φis homogeneous (or
stationary), if TxΦhas the same distribution as Φfor
allx∈R2. The law of a homogeneous tessellation is
alsocalledhomogeneous.
Wefixnowtwo randomplanartessellations Φand
Ψwith law PandQ, respectively. The cells of Φ
are denoted by C(Φ). We associate now to each cell
p∈C(Φ)independentlyarandomtessellation Ψpwith
lawQ. Nowdefinea randomtessellation Φ⊚Ψby
Φ⊚Ψ:=Φ∪/uniondisplay
p∈C(Φ)(p∩Ψp),
the law of which is denoted by P⊞Q. It was shown
in Mecke et al.(2008b) that if Φis a homogeneous
tessellation and Qis also homogeneous then Φ⊚Ψ
is a homogeneous random tessellation, too. We call
a homogeneous random tessellation Φstable with
respect to iteration (STIT for short), if 2 (Φ⊚Φ)has
the same law as Φitself (this is equivalent to the
definitionusedforexampleinNagelandWeiss(2005),
seeMecke,2008).
The existence of such tessellations was shown in
NagelandWeiss(2005)togetherwithaconstructionin
a bounded window for arbitrary dimensions. A global
construction of planar STIT tessellations was recently
presentedin Mecke etal.(2008a).
Denote by [H,H]the measurable space of lines
through the origin. By the direction of an arbitrary
linegwe mean the unique parallel line r(g)∈H.
For a line segment sdenote the line gcontaining s
byg(s). Then the direction r(s)ofscan be defined
asr(s):=r(g(s))∈H. For a planar tessellation Φ
we introduce now a directional measure κas follows:
ForB∈Hwe consider the family ΦBof edges with
direction in B. The mean length of these edges per
unit area (note that this is a well defined quantity,
sinceΦis homogeneous) will be denoted by LA(B).
By the relation κ:H→[0,∞):B/mapsto→LA(B)we get a
measureκon[H,H], the directional measure of Φ.
If we denote by LAthe edge length intensity, i.e., the
mean total edge length per unit area, of Φ, we can
writeκalsoasκ=LAϑforsomeprobabilitymeasureϑon[H,H], sinceκ(H) =LA. The latter is called
directionaldistribution of Φ. We assumein this paper,
thatκ(orequivalently ϑ)isconcentratedonmorethan
a single direction. This ensures the existence of of a
STIT tessellation with this measure as its directional
measure or distribution ( cf.Nagel and Weiss, 2005).
Having in mind these notions and notations we can
definetheroseofintersectionsof κas
sκ:H→(0,∞):h/mapsto→/integraldisplay
H|sin∠(h,˜h)|dκ(˜h).(1)
Fig.1.Realizationofan isotropicSTITtessellation.
We summarize now the most important features
of planar STIT tessellations Φwith directional
measureκ:
1. The intersection of Φwith an arbitrary line g(not
necessarily through the origin) is a homogeneous
Poisson point process on g. It has intensity
sκ(r(g))(cf.NagelandWeiss,2005).
2. The interior of the typical cell (this is a cell with
Palm shape distribution, see Stoyan et al., 1995)
has the same distribution as the interior of the
typical cell of a Poisson line tessellation with the
same directional measure ( cf.Nagel and Weiss,
2003).
3. The nodes (vertices) have T-shape,i.e., from each
node we have three emanating edges and two of
themarecollinear( cf.NagelandWeiss,2003).
4. The cells of Φare not in a face-to-face position,
seeFig.1 ( cf.NagelandWeiss,2005).
More results about STIT tessellations can be found
in Nagel and Weiss (2003), Nagel and Weiss (2005),
NagelandWeiss (2006),Mecke etal.(2007),Mecke et
al.(2008a),Mecke etal.(2008b),Th¨ ale(2008),Th¨ ale
(2009).
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ImageAnalStereol2009;28:69-76
RECENTRESULTSFOR I-,K-
ANDJ-SEGMENTS
MackisackandMiles(1996)introducedthenotion
ofI-,K-andJ-segmentsandshowedthattheiranalysis
can be fruitful, especially in the case of tessellations
which are not face-to-face. A K-segment they called
every line segment of the tessellation without any
vertex in its relative interior. A J-Segment is a face
of a cell and an I-segment is the union of connected
and collinear K-segments, which cannot be enlarged
by another K-segment.
Fig. 2.Different types of linear segments in a planar
tessellation.
Theinvestigationofthelengthdistributionofthese
segmentsforplanarSTIT tessellationsstarts in Mecke
et al.(2007) for the isotropic case.The authors of this
paper were able to calculate explicitly the densities
of the length distribution of the segments and by
integration their mean values and second moments
(comparewith the isotropicexamplebelow).
In his paper, Mecke (2008) starts the analysis of
the anisotropic case. Here, in contrast to the isotropic
case, one has to take also into accountthe direction of
thesegments.Thus,heconsideredthejointdistribution
of direction andlength for the case of I-segments.
His approach was completed by the consideration of
the typical K- andJ-segment in Th¨ ale (2008) (here
typicalshouldbe understoodin the Palmsense,where
the Palm distribution with respect to the segment
midpoints isconsidered).
Weliketosummarizetheresultsnow.Lettherefore
Φbe a homogeneous planar STIT tessellation with
directional measure κ. The rose of intersections of κ
is againgivenby Eq. 1. The directionaldistribution of
Φwillbedenotedby ϑ.Wefurtherdefinetheconstant
ζκ:=/integraldisplay
H/integraldisplay
H|sin∠(h,˜h)|dκ(h)dκ(˜h).
The joint distribution of direction and length of the
typicalX-segment, X∈ {I,J,K}, is a probabilitymeasureon H×(0,∞)havingdensity dXwithrespect
to the product measure κ×L+, where L+is the
Lebesgue measure restricted to (0,∞). Then we have
(Theorem 12 in Mecke, 2008, Corollary 3.4 and
Corollary 3.6in Th¨ ale,2008):
dI:(h,x)/mapsto→2
ζκsκ(h)/integraldisplaysκ(h)
0t2e−txdt,
dK:(h,x)/mapsto→2
3ζκs2
κ(h)/integraldisplay2
1t2e−sκ(h)txdt,
dJ:(h,x)/mapsto→sκ(h)
ζκ/integraldisplay∞
xe−sκ(h)tdt.
We consider now the marked point process αX=
{yk,hk}k∈N,X∈ {I,J,K}, ofX-segmentmidpoints yk,
where the marks hk∈Hare given by the direction
of theX-segment through yk. The typical X-segment
in direction h∈Hof the tessellation Φcan now
be defined as the line segment containing the origin,
whereΦis considered under the Palm distribution
P0,h. (For the general theory of Palm distributions for
marked point processessee for example Stoyan et al.,
1995).
From the formulas above we can conclude the
following integral representations for the conditional
densities dI|h,dJ|handdK|hof the length distributions
ofthetypical I-,K-andJ-segmentindirection h∈H,
since the density of the direction with respect to the
directional measure κof the typical I-,K- andJ-
segment is given by sκ(·)/ζκin each case ( cf.Mecke,
2008;Th¨ ale,2008):
dI|h(x) =2
s2κ(h)/integraldisplaysκ(h)
0t2e−txdt, (2)
dK|h(x) =2
3sκ(h)/integraldisplay2
1t2e−sκ(h)txdt,(3)
dJ|h(x) =/integraldisplay∞
xe−sκ(h)tdt. (4)
The densities written in this form (and not
explicitly without an integral) will turn out to be very
powerfulandleadto shortproofin thenextsection.
MAINRESULTS
We consider a homogeneous planar STIT
tessellation Φwith directional measure κ. Denote the
rose of intersections of κagain by sκand fix some
direction h∈H. The typical I-,K- andJ-segment
with direction hwill be denoted by I0,h,K0,hand
J0,h, respectively. The (euclidean) length of I0,h,K0,h
andJ0,his denoted by L(I0,h),L(K0,h)andL(J0,h),
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TH¨ALEC:Momentsoflengthof linesegmentsinhomogeneousplanarSTI Ttessellations
respectively. For k=1,2,3,...we define now the k-
moments
Ik(h):=E|L(I0,h)|k,
Kk(h):=E|L(K0,h)|k,
Jk(h):=E|L(J0,h)|k.
TYPICAL I-SEGMENTS
We start our analysis with the typical I-segment
I0,hin direction h∈H. First observe that integration
byparts yieldsfor t>0
/integraldisplay∞
0xke−txdx=t−k−1Γ(k+1),
whereΓ(·)is Euler’s Gamma-function. We can now
useFubini’stheoremandformula Eq.2to obtain
Ik(h) =/integraldisplay∞
0xkdI|h(x)dx
=/integraldisplay∞
0xk/bracketleftbigg2
sκ(h)2/integraldisplaysκ(h)
0t2e−txdt/bracketrightbigg
dx
=2
sκ(h)2/integraldisplaysκ(h)
0t2/bracketleftbigg/integraldisplay∞
0xke−txdx/bracketrightbigg
dt
=2
sκ(h)2Γ(k+1)/integraldisplaysκ(h)
0t2t−k−1dt
=2
sκ(h)2Γ(k+1)/integraldisplaysκ(h)
0t−k+1dt
=

2
sκ(h):k=1,
+∞:k≥2.
This means for the length of the typical I-segment in
any direction h, that only the first moment exists. All
highermomentsareinfinite.
TYPICAL K-SEGMENTS
We continue with the typical K-segment K0,hwith
direction h∈H. Firstobservethat
/integraldisplay∞
0xke−stxdx= (st)−k−1Γ(k+1)
for alls,t>0. We use now formula Eq. 3 and again
Fubini’stheorem toobtainKk(h) =/integraldisplay∞
0xkdK|h(x)dx
=/integraldisplay∞
0xk/bracketleftbigg2
3sκ(h)/integraldisplay2
1t2e−sκ(h)txdt/bracketrightbigg
dx
=2
3sκ(h)/integraldisplay2
1t2/bracketleftbigg/integraldisplay∞
0xke−sκ(h)xdx/bracketrightbigg
dt
=2
3sκ(h)−kΓ(k+1)/integraldisplay2
1t2t−k−1dt
=2
3sκ(h)kΓ(k+1)/integraldisplay2
1t−k+1dt
=

2
3sκ(h)kΓ(k+1)2k−4
2k(k−2):k/ne}ationslash=2,
4
3sκ(h)2ln2 : k=2.
This shows, that in contrast to the previous case all
moments of the length of the typical K-segment with
direction hexist.Wealsoobtainedaclosedformulafor
them.
TYPICAL J-SEGMENTS
This case is much easier, since L(J0,h)is
exponentially distributed with parameter sκ(h),
because of the Poisson typical cells of STIT
tessellations (see the key properties in the section on
planarSTITtessellations).Thus,we obtain
Jk(h) =E|L(J0,h)|k=Γ(k+1)
sκ(h)k.
We see that also in the case of the typical J-
segment in direction h allmoments exist. We like to
remark that the result can also be obtained by a direct
calculationusingformula Eq.4.
Note that in the above formulas for Ik(h),Kk(h)
andJk(h),Γ(k+1)couldbereplacedby k!.
MEAN VALUES AND VARIANCES
Wecanespeciallyapplyourresultstocomputethe
mean values and the variances of L(I0,h),L(K0,h)and
L(J0,h). We consider therefore a homogeneous planar
STIT tessellation with directional measure κand fix a
direction h∈H. Thenweobtain
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ImageAnalStereol2009;28:69-76
EL(I0,h) =2
sκ(h),
VL(I0,h) = +∞,
EL(K0,h) =2
3sκ(h),
VL(K0,h) =2
3sκ(h)2/parenleftbigg
2ln2−1
3/parenrightbigg
,
EL(J0,h) =1
sκ(h),
VL(J0,h) =1
sκ(h)2,
by using the formula VX=EX2−(EX)2for real
valuedrandomvariables X.
MOMENT RELATIONSHIPS
Recall that sκ(·)/ζκis the density of the direction
of the typical I-,J- andK-segmentsegment I0,K0,J0
wrt.κ(cf.Mecke,2008; Th¨ ale, 2008)and the relation
κ(H) =LA.We obtainthe following meanvaluesfor
the lengthofthetypical I-,J-andK-segment:
EL(I0) =Eh(EL(I0,h)) =2LA
ζκ,(5)
EL(K0) =Eh(EL(K0,h)) =2LA
3ζκ,(6)
EL(J0) =Eh(EL(J0,h)) =LA
ζκ, (7)
where Ehdenotestheaverageoveralldirections h.For
example
EL(I0) =/integraldisplay
H2
sκ(h)·sκ(h)
ζκdκ(h) =2
ζκκ(H) =2LA
ζκ.
Note that these mean values are the same as the one
computed in Nagel and Weiss (2006), but there the
constantζ=ζκ/L2
Awasusedinsteadof ζκ.
These formulas imply the following mean value
relations:
EL(I0,h) =3EL(K0,h) =2EL(J0,h)
and
EL(I0) =3EL(K0) =2EL(J0).(8)
The relations Eq. 8 confirm earlier results from the
paper Nagel and Weiss (2006), which were obtained
by quite different methods. Beside these mean value
relations we obtain the following relationship of
the variances of the typical K- andJ-segment with
direction h∈H:
VL(K0,h) =2
3/parenleftbigg
2ln2−1
3/parenrightbigg
VL(J0,h)
≈0.70196 ·VL(J0,h).For the higher moments Kk(h)andJk(h),k≥3, we
get the following general relation for the conditional
lengthdistributions:
Kk(h) =2
3·2k−4
2k(k−2)Jk(h)
for anyh∈H. An analogous formula can also be
obtained for the unconditionedmoments again simply
byintegration:
E|L(K0)|k=2
3·2k−4
2k(k−2)E|L(J0)|k.
Notethat
lim
k→∞2
3·2k−4
2k(k−2)=0.
Comment: One can conclude from Eq. 8 the fact,
that the mean number of nodes in the relative interior
ofthetypical I-segmentequals2and 1/2forthetypical
J-segment. It will be shown in the forthcoming paper
(Th¨ ale,2009),thatforSTITtessellationsin Rd,d≥2,
the mean number of nodes in the relative interior of
the typical I-segment equals dand(d−1)/2 for the
typicalJ-segment ( I-,J- andK-segments of higher
dimensionalSTITtessellationsarenowconsideredfor
the 1-skeleton;the definitionremainsthesame).
EXAMPLES
THEISOTROPICCASE
We consider in this section homogeneous and
isotropic planar STIT tessellations with edge length
intensity π/2. In this case we can calculate the
distribution functions FI,FKandFJof the length of
the typical I-,K- andJ-segment, respectively, using
formulas Eqs.2–4.Weobtain
FI(x) =1−2
x2/parenleftbig
1−(1+x)e−x/parenrightbig
,
FK(x) =1−2
3x2/parenleftbig
1+x−(1+2x)e−x/parenrightbig
e−x,
FJ(x) =1−e−x,
which confirms the results of Mecke et al.(2007),
especially Theorem 2 and 4 therein. For the densities
pI(x),pK(x)andpJ(x)weget
pI(x) =4
x3/parenleftbigg
1−/parenleftbig
1+x+x2
2/parenrightbig
e−x/parenrightbigg
,
pK(x) =4
3x3/parenleftbigg/parenleftbig
1+x+x2
2/parenrightbig
e−x
−/parenleftbig
1+2x+2x2/parenrightbig
e−2x/parenrightbigg
,
pJ(x) =e−x.
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TH¨ALEC:Momentsoflengthof linesegmentsinhomogeneousplanarSTI Ttessellations
Fig.3.Densities p I(black), p K(grey)and p J(dashed)
in theisotropiccase.
For the mean values and varianceswe have in this
case
EL(I0) =2,EL(K0) =2
3,EL(J0) =1
and
VL(I0) = +∞,
VL(K0) =4
3/parenleftbigg
2ln2−1
3/parenrightbigg
,
VL(J0) =1
from Eqs. 5–7, which confirms Eq. 8 and the earlier
resultsfrom NagelandWeiss (2006).
Note that in the isotropic case the length and
the direction of the typical I-,K- andJ-segment are
independent.
THERECTANGULAR CASE
We consider a homogeneous planar STIT
tessellation Φwith directionaldistribution
ϑ(h) =

1
2:histhex-ory-axis,
0 : else ,
and edge length intensity 1. The cells of such a
tessellation Φare rectangles with probability one (a
similar case was investigated in Mackisack and Miles
(1996)butnotforSTIT tessellationsofcourse).
Fig.4.Realizationofa rectangularSTITtessellation.
From Eqs. 2–4 we calculate again the densities
pI(x),pK(x),pJ(x)of the length distribution of the
typicalI-,K- andJ-segmentwith direction in x- ory-
axis,respectively:
pI(x) =2
x3/parenleftBig
8−(8+4x+x2)e−x
2/parenrightBig
,
pK(x) =2
3x3/parenleftBig
8+4x+x2−e−x
2/parenleftbig
8+8x+4x2/parenrightbig/parenrightBig
e−x
2,
pJ(x) =1
2e−x
2.
Fig.5.Densities p I(black), p K(grey)and p J(dashed)
in therectangularcase.
Forthemeanvaluesandvariancesweobtain
EL(I0,h) =4,EL(K0,h) =4
3,EL(J0,h) =2
and
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ImageAnalStereol2009;28:69-76
VL(I0,h) = +∞,
VL(K0,h) =16
3/parenleftbigg
2ln2−1
3/parenrightbigg
,
VL(J0,h) =4
for bothdirections h(xandy). Furthermorewe have
EL(I0) =1
2(EL(I0,x)+EL(I0,y)) =4,
EL(K0) =1
2(EL(K0,x)+EL(K0,y)) =4
3,
EL(J0) =1
2(EL(J0,x)+EL(J0,y)) =2.
ACASE WITH UNEQUALWEIGHTS
At the end we like to discuss an example with
unequal weights. Consider therefore a directional
distribution ϑ, which has the following weights on
three directions
ϑ(h) =

1
2:h=x-axis,
1
3:h={x=y},thediagonal ,
1
6:h=y-axis.
The first direction is called D1, the second D2
and the third D3for abbreviation. We further assume
that the edge length intensity LAis 1. On can now
calculate the 9 different densities pX,Di(·)forX∈
{I,J,K}andi=1,2,3, butwe do notgivethe explicit
formulashere.Insteadweconcentrateonseveralmean
value relations: For the mean values and variances in
direction D1wehave
EL(I0,D1) =4,EL(K0,D1) =4
3,EL(J0,D1) =2
and
VL(I0,D1) = +∞,
VL(K0,D1) =16
3/parenleftbigg
ln2−1
3/parenrightbigg
,
VL(J0,D1) =4.
This arethesamevaluesasin therectangularcase.
For the mean values and variances in direction D2
we get
EL(I0,D2) =6,EL(K0,D2) =2,EL(J0,D2) =3and
VL(I0,D2) = +∞,
VL(K0,D2) =4(6ln2−1),
VL(J0,D2) =9.
For the mean values and variances in direction D3
weobtain
EL(I0,D3) =12,EL(K0,D3) =4,EL(J0,D3) =6
and
VL(I0,D3) = +∞,
VL(K0,D3) =16(6ln2−1),
VL(J0,D3) =36.
Here we see that the mean values in direction D2are
half of the mean values in direction D3. This is due
to the fact that direction D2is twice more likely to
occur for an edge than direction D3. Note that there is
asimilarrelationbetweenthemeanvaluesintheother
directions.
For the mean length of the typical I-,K- andJ-
segmentI0,K0andJ0weobtainnow
EL(I0) =1
2EL(I0,D1)+1
3EL(I0,D2)+1
6EL(I0,D3)
=6,
EL(K0) =1
2EL(K0,D1)+1
3EL(K0,D2)+1
6EL(K0,D3)
=2,
EL(J0) =1
2EL(J0,D1)+1
3EL(J0,D2)+1
6EL(J0,D3)
=3,
whichconfirmstherelation in Eq.8.
ACKNOWLEDGMENTS
The author would like to thank Joachim Ohser
(Darmstadt) for providing the beautiful pictures of
the STIT tessellations, Werner Nagel (Jena) for his
helpful comments and interesting discussions and the
refereesforimprovingconsiderablythepresentationof
the manuscript.
This work was supported by the Schweizerischer
NationalfondsgrantSNFPP002-114715/1.
75
TH¨ALEC:Momentsoflengthof linesegmentsinhomogeneousplanarSTI Ttessellations
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