Image Anal Stereol 2014;33:55-64 doi:10.5566/ias.v33.p55-64 Original Research Paper PLANAR SECTIONS THROUGH THREE-DIMENSIONAL LINE-SEGMENT PROCESSES SASCHA DJAMAL MATTHESC,1 AND DIETRICH STOYAN2 1Institut f¨at Bergakademie Freiberg, D-09596 ur Keramik, Glas-und Baustofftechnik, Technische Universit¨Freiberg, Germany; 2Institut f¨at Bergakademie Freiberg, D-09596 Freiberg, ur Stochastik, Technische Universit¨ Germany e-mail: Sascha-Djamal.Matthes@ikgb.tu-freiberg.de (Received September 25, 2013; revised January 14, 2014; accepted January 14, 2014) ABSTRACT This paper studies three–dimensional segment processes in the framework of stochastic geometry. The main objective is to .nd relations between the characteristics of segment processes such as orientation-and length-distribution, and characteristics of their sections with planes. Formulae are derived for the distribution of segment lengths on both sides of the section plane and corresponding orientations, where it is permitted that there are correlations between the angles and lengths of the line-segments. Keywords: .bre process, .bre-reinforced materials, line-segment process, stereology, stochastic geometry. INTRODUCTION Segment processes are stochastic models for random systems of line-segments randomly scattered in space. They belong to the more general class of .bre processes, the mathematical theory of which was developed by Joseph Mecke and coworkers (Mecke and Nagel, 1980; Mecke and Stoyan, 1980b; Chiu et al., 2013). These processes .nd important applications in the context of .bre-reinforced materials, where .bres, which can be often modelled as line-segments of negligible thickness, are embedded in a matrix of more or less homogeneous material. In the now classical papers mentioned above planar sections played an important role. Such sections produce systems of .bre–plane intersection points, which can be statistically analysed with the aim to get information on the spatial .bre system. This setting belongs to the .eld of stereology, and a classical formula there is LV = 2NA , (1) where LV is the mean total .bre length per unit volume and NA the number of intersection points per unit area. (The formula holds true under the assumptions of statistical homogeneity or stationarity and isotropy, see also Chiu et al., 2013.) Planar sections through segment processes appear in the context of .bre-reinforced materials, when axial tension is studied. Following Li et al. (1991) the intersections of .bres with a plane orthogonal to the tension axis are investigated. Additionally to the characteristics studied when stereology is of interest, also the residual lengths of the line-segments on both sides of the section plane are of importance in the mechanical calculations. (They have never been considered in the stereological context, since these lengths cannot be measured in the section plane.) The present paper .rst explains a natural segment process model, following the pattern of Mecke and Stoyan (1980a). Then it derives formulae for the section process characteristics. Some of them have counterparts in the classical theory, while those related to the residual lengths are new, generalising results in Li et al. (1991), who considered the case of segments of constant length. Furthermore, formulae for the maximum and minimum residual segment length are derived since these characteristics play a role in calculations of the contribution of .bres to the mechanical strength in a composite material. MODEL DESCRIPTION This paper considers three–dimensional line-segment processes. A realisation of such a process is a set of randomly distributed line-segments in space. To characterise such a line-segment we use its top point (in the sense of the z-axis) (x,y,z) . R3, the length l > 0 and the angles . and ß denoting azimuth and polar angle of the line-segment. Since we are not interested in the sense of direction of the line-segment, d . i with . . [0,2.] and ß . 0, a line-segment is well 2 de.ned. marked point process .V (for more on marked point processes see Chiu et al., 2013). Realisations of .V can be written as sequences of marked points: .V = {[(xi,yi,zi),lV ,.V ,ßV ]} , (2) ii i with (xi,yi,zi) . R3 ,lV > 0,.V . [0,2.] and ßV . di ii i 0, . . Moreover we introduce the stochastic variables 2 of the polar angle, BV , the azimuthal angle, .V , and the .bre length, LV , of a typical line-segment of .V . Here and in the following we assume .V to be stationary, i.e., the distribution of .V is translation invariant. It is not assumed that the marked point process .V has some speci.c distribution, e.g.,a marked Poisson process. The results presented in this paper hold for every distribution of a stationary .V , where BV and .V are stochastically dependent (see page 57) or independent (see page 60) of LV . ESSENTIAL PROPERTIES The distribution of the marked point process .V is described by the following characteristics: Table 1. Characteristics of the spatial marked point process .V NV mean number of top points of line-segments per unit volume FV,L,B,.(l,ß, .) joint distribution function of .bre length LV and angles BV and .V of a typical line segment FV,B(ß), FV,.(.), marginal distribution functions FV,L(l) of the stochastic variables BV , .V and LV . The stochastic variables of the polar angle, BV , and the azimuthal angle, .V may depend on the stochastic variable of the spatial .bre length, LV . In order to study the mechanical behavior of .bre-reinforced materials under axial tensions the intersection of a plane with a line-segment system is of peculiar interest. The mechanical effect of a .bre in a homogeneous material depends on the intersection angle and the length of the .bre under and over the plane respectively. Thus these quantities have to be studied. This approach appears in the classical papers by Li et al. (1991), Brandt (1985) and in subsequent work. However, in these papers the segment lengths are assumed to be constant. Due to our homogeneity assumption we choose the intersecting plane to be the (x,y)-plane S = {(x,y,z) . R3: z = 0}. Intersection of the segments of .V with Image Anal Stereol 2014;33:55-64 S yields the marked point process .A of intersection points with the realizations AA .A = {[(.i,.i),r2,i,. A ,ßiA]} , (3)1,i,ri AA where (.i,.i) are the intersection points, r1,i and r 2,i the lengths of the upper and lower part of the segments respectively and .iA and ß A the corresponding section i angles. Due to stationarity of .V also the marked point process .A is stationary. We introduce the stochastic variables of the polar section angle, BA , the azimuthal section angle .A and the upper and lower part of the segment which belongs to the typical intersection point, RA and 1 RA respectively. The basic constants and distribution 2 functions of the marked point process .A are shown in the following table: Table 2. Characteristics of the planar marked point process .A NA mean number of section points of .A per unit area FA,R1,R2,.,B(r1,r2,.,ß) joint distribution function of upper and lower segment lengths and intersection angles BA , .A at a typical section point FA,R1,R2 (r1,r2), marginal distribution FA,.(.), FA,B(ß) functions of the upper and lower segment lengths and the intersection angles BA and .A . BASIC CHARACTERISTICS OF .A AND RELATIONS TO .V MAIN RESULTS The main objective at this point is to establish relations between the basic constants and distribution functions of .V and those of .A. Due to the choice of the intersection plane S and the de.nition of the azimuthal angle ., the latter can be ignored. We concentrate on relations between NA, NV and the marginal distribution functions FA,R1,R2,B(r1,r2,ß) and FV,L,B(l,ß). The following general basic equation di holds for NA,NV > 0, r1,r2 > 0 and ß . 0, . 2: NAFA,R1,R2,B(r1,r2,ß)= min{r1,r2} hß h = NV sinß' FV,L,B(l + max{r1,r2},ß') 00 - FV,L,B(l,ß')dl dß' minh{r1,r2} + NV cosß FV,L,B(l + max{r1,r2},ß) 0 - FV,L,B(l,ß)dl. (4) The relation of the corresponding probability density functions fV,L,B(l,ß ) and fA,R1,R2,B(r1,r2,ß) is therefore NA fA,R1,R2,B(r1,r2,ß)= NV cosß fV,L,B(r1 + r2,ß) . (5) Eq. 4 is the starting point for some important formulae. Let ß = . 2 and r1,r2 › .. Then we obtain the intensity of .A as NA = NV E(LV cosBV ) , (6) where h. 2 h. E(LV cosBV )= l cosß fV,L,B(l,ß )dl dß . 00 This expression simpli.es for the isotropic case and if LV and BV are stochastic independent, see Eq. 26. With the latter relation we are able to determine FA,R1,R2,B if NV and FV,L,B are given. Furthermore, we obtain with Eq. 4, Eq. 6 and ß = . the joint distribution 2 function . FA,R1,R2 (r1,r2)= FA,R1,R2,Br1,r2, 2 of the upper and lower segment lengths. These distributions exist if EcosBV = 0, i.e., if the case of all .bres parallel to the section plane is excluded. NAFA,R1,R2 (r1,r2)= min{r1,r2} hh. 2 = NV sinß'FV,L,B(l + max{r1,r2},ß ') 00 - FV,L,B(l,ß')dß'dl , FA,R1,R2 (r1,r2)= min{r1,r2} hh. 2 = 1 sinß'· E(LV cosBV ) 00 FV,L,B(l + max{r1,r2},ß ') - FV,L,B(l,ß')dß 'dl , (7) with 0 . r1,r2 < .. We can determine the marginal distribution functions for the upper and lower segment lengths FA,R1 (r1)= lim FA,R1,R2 (r1,r2) r2›. and FA,R2 (r2)= lim FA,R1,R2 (r1,r2). r1›. The stochastic variables RA 1 and RA 2 are distributed with the distribution function FA,R(r)= FA,R1 (r)= FA,R2 (r)= . hr h2 1 ' identically sinßFV,B(ß' )E(LV cosBV ) 00 - FV,L,B(l,ß' ) dß ' dl , (8) for r > 0 and FV,B(ß)= lim FV,L,B(l,ß). l›. With Eqs. 6 and 4 and r1,r2 › . we analogously get the distribution function of the section angle BA FA,B(ß)= h. hß ' ' 1 sin ßFV,B(ß' ) - FV,L,B(l,ß' ) dßE(LV cosBV ) 0 0 + cosß FV,B(ß) - FV,L,B(l,ß)dl . (9) With Eqs. 6–9 we have explicit relations for the basic characteristics of .A. We can add also the probability density functions using Eqs. 6–9: fA,R1,R2 (r1,r2)= . h2 1 cosß fV,L,B(r1 + r2,ß )dß , (10)E(LV cosBV ) 0 and fA,B(ß)= h. h. 1 cosß fV,L,B(r1 + r2,ß )dr1dr2 . E(LV cosBV ) 00 (11) The length of the line-segments were assumed to be stochastically dependent on the angle BV throughout the above investigations. Eq. 4 shows that, consequently, the line-segment lengths RA 1 and RA are 2 stochastically dependent of the intersection angle BA . SKETCH OF THE PROOF OF EQ. 4 The proof follows the pattern of Mecke and Stoyan (1980a), which is also used in Mecke and Stoyan (1980c). Let SD(t1,t2,b,c) be the expected number of segments pi = [(xi,yi,zi),liV ,.V ,ßV ] of the marked ii point process .V ful.lling the following conditions: 1. the line-segment pi intersects a given compact subset D of the plane S, 2. the line-segment has intersection angles ßA . [0,b] i with b . 0, . and .A . [0, c] with c . [0,2.], d i 2 i 3. the lengths of the segment above and below S ful.l AA r1,i . t1 and r2,i . t2 with t1,t2 > 0. The quantity SD(t1,t2,b,c) can be calculated in terms of .A and .V , and equating the corresponding terms yields Eq. 4. The Campbell theorem (see, e.g., Chiu et al., 2013) applied to .A yields simply SD(t1,t2,b,c)= NAFA,R1,R2,B,.(t1,t2,b,c) (12) d i for t1,t2 > 0, b . 0, . and c . [0,2.]. 2 In terms of .V , SD(t1,t2,b,c) can be expressed as SD(t1,t2,b,c)= E. f (pi,t1,t2,b,c)(13) pi..V d i for t1,t2 > 0, b . 0, . and c . [0,2.], pi = 2 [(xi,yi,zi),liV ,.V ,ßV ] denotes a single segment and f ii is the indicator function with f (pi,t1,t2,b,c)= 1 for segments that ful.l the conditions 1–3 of the de.nition of SD, otherwise f = 0. In the following we describe the function f . AA = lV With r1,i + rit follows 0 . lV . t1 + t2 as 2,ii i a .rst constraint. Furthermore we have ßV . [0,b] and i .V . [0,c], therefore we choose i f (pi,t1,t2, b,c)= 1[0,t1+t2](liV )1[0,b](ßV )1[0,c](.V )1Z0 (xi,yi,zi) ii for some set Z0 = Z0(.V ,ßV ,liV ,t1,t2) with (xi,yi,zi) . ii Z0 if and only if pi ful.ls the conditions 1 and 3. Let Z1 = D .s · ,s . [0, min{lV ,t1}] e.V ,ßVi ii and Z2 = D .s · e.V ,s . [lV - min{lV ,t2},liV ] i ,ßiVi i Image Anal Stereol 2014;33:55-64 with e.,ß =(cos. sinß,sin. sinß, cosß), where . denotes the Minkowski addition. Then pi intersects the A set D with an upper segment length r1,i . t1 if and only if (xi,yi,zi) . Z1 and pi intersects D with a lower A segment length r2,i . t2 if and only if (xi,yi.zi) . Z2. In both cases pi intersects D with the intersection angles ßiV . [0,b] and .iV . [0,c]. It follows that pi ful.ls the conditions 1 to 3 if and only if (xi,yi,zi) . Z0 = Z1 .Z2. Due to the structure of Z1 and Z2 we can write Z0 as the Minkowski sum of D and a line-segment: Z0 = to the representation of Z0 as a Minkowski sum we can express this integral by the product of the area of D, .2(D), and the height of Z0. We get h 1Z0 (x,y,z)d(x,y,z)= R3 .2(D)cosß min{l,t1} + min{l,t2}- l . At this point the integrand is independent of .. With Eq. 12 it follows the equivalence of the marginal distributions D . s · e.V ,s . [lV - min{t2,liV },min{t1,liV }] . i ,ßiV i Fig. 3 explains the underlying geometry. Figure 3. Underlying geometry in the (x,z)-plane of the intersection of a segment of azimuthal angle .V = 0 i with the plane S. The right hand side of Eq. 13 can be calculated by means of the Campbell theorem. We obtain SD(t1,t2,b,c)= hh = NV 1Z0 (x,y,z)· R3 [0,t1+t2]×[0,b]×[0,c] dFV,L,B,.(l,ß,.)d(x,y,z) h t1+t2h hb hc = NV 1Z0 (x,y,z) R3 0 0 0 fV,L,B,.(l,ß,.)d. dß dl d(x,y,z) . (14) FA,.(.) . FV,.(.) . (15) Hence follows, we concentrate on the relation of the marginal distribution functions FA,R1,R2,B and FV,L,B and SD(t1,t2,b,2.). We obtain SD(t1,t2,b,2.)= h NV cosß· [0,t1+t2]×[0,b] min{l,t1} + min{l,t2}- l dFV,L,B(l,ß) . (16) Equating Eq. 16 with Eq. 12 and applying integration by parts gives the desired relation Eq. 4. APPLICATIONS AND DISCUSSION The stereological formulas Eqs. 6–9 can now be used to verify earlier results and to examine application-related cases. SUPERPOSITION OF LINE-SEGMENT PROCESSES In the following example the line-segments follow a special relation between length and direction: long .bres (l . [l0,lmax],0 < l0 < lmax) all have the same di polar angle ß0 . 0, . while short .bres (l . [0,l0]) 2 are isotropic. The proportion of segments of length l . [0,l0] is p . [0,1] and for segments with l . [l0,lmax] is 1 - p. The joint probability density function of LV and BV is therefore p . sinß , l . l0 . l0 fV,L,B(l,ß)= 1-p .ß0 (ß) , l0 . l . lmax . lmax-l0 0 , l > lmax. (17) Thus, this example represents a superposition of two different line-segment processes. With Eq. 6 the mean By applying Fubini’s theorem we can rearrange number of section points per unit area is the order of integration, and we evaluate 1Z0 (x,y,z)d(x,y,z) using Cavalieri’s principle. Due NA = NV1 pl0 + 1 cos ß0(1 - p)(lmax + l0). (18) R3 42 Moreover using Eq. 7 and Eq. 8 the conditional distribution function FA,R1|R2.r2 (r1)= . min{r1,r2} hh2 ' sinßFV,L,B(l + max{r1,r2},ß' ) 00 - FV,L,B(l,ß' ) dß'dl . hr2 h2 ' sin ßFV,B(ß' ) - FV,L,B(l,ß' ) dß' dl 00 as well as the conditional density function fA,R1|R2.r2 (r1)= . 2 ' ' sinß FV,L,B(r1 + r2,ß' ) - FV,L,B(r1,ß) dß0 . r22 ' sinß FV,B(ß ' ) - FV,L,B(l,ß ' ) dß 'dl 00 can be calculated. The conditional probability density function fA,R1|R2.r2 (r1) is shown in Fig. 4 for l0 = 1, 33 lmax = 3, ß0 = and p = and by means of Eq. 18 the 44 ratio NNVA is 0.553. Thus, NA can be determined once NV is known. Figure 4. The conditional probability density function fA,R1|R2.r2 (r1) of the upper segment length R1 A under the condition the lower segment length RA 2 is bounded from above. INDEPENDENT FIBRE ANGLES AND LENGTHS Assume the line-segment length LV is stochastically independent of the angles .V and BV for the line-segment process .V . Under this assumption this line-segment process can be characterized using the following parameters and functions: Table 3. Characteristics of the spatial marked point process .V under the assumption that LV and BV are stochastically independent. NV mean number of top points of line segments per unit volume FV,L(l) distribution function of length of a typical segment of .V FV,.(.), FV,B(ß ) distribution function of azimuth and polar angle of a typical segment of .V Eq. 4 simpli.es in the following way: NAFA,R1,R2,B(r1,r2,ß)= minh{r1,r2}hß = NV FV,L(l + max{r1, r2}) - FV,L(l) 00 · cosß' dFV,B(ß' )dl min{r1,r2} h = NV FV,L(l + max{r1,r2}) - FV,L(l) dl 0 hß ' · sinß'FV,B(ß' ) dß 0 for r1, r2 > 0, ß . d 0, . i and NA,NV > 0. It follows 2 that the segment lengths RA 1, RA and the polar angle 2 BA of the segment process .A are stochastically independent. We therefore concentrate on the marginal distribution functions FA,R1,R2 (r1,r2) and FA,B(ß). We have NAFA,R1,R2 (r1,r2)FA,B(ß)= min{r1,r2} h NV FV,L(l + max{r1,r2}) - FV,L(l) dl· 0 hß ' sinß' FV,B(ß ' )dß. (19) 0 Using this simpli.ed relation we obtain NA = NV ELV EcosBV , (20) FA,R1,R2 (r1,r2)= min{r1,r2} h 1 FV,L(l + max{r1,r2}) - FV,L(l) dl ELV 0 (21) Image Anal Stereol 2014;33:55-64 and ' FA,B(ß)= 1 hß sinß' FV,B(ß' ) dß . (22) EcosBV 0 The segment lengths RA and RA are identically 12 distributed, i.e., the marginal distributions FA,R1 and FA,R2 are equivalent: FA,R(r)= FA,R1 (r)= FA,R2 (r)= hr 1 1 - FV,L(l) dl , r > 0 . (23) ELV 0 Concerning the corresponding probability density functions of .V and .A it holds 1 fA,R1,R2 (r1,r2)= fV,L(r1 + r2) , (24) ELV and 1 fA,B(ß)= fV,B(ß)cosß . (25) EcosBV THE ISOTROPIC CASE We assume that the line-segment length LV is stochastically independent of the angles .V and BV for the line-segment process .V . Furthermore the line-segment process .V is assumed to be isotropic, i.e., the directional vector of the typical line-segment is uniformly distributed on the unit sphere. We have fV,B(ß)= sinß , and the distribution function FV,B(ß)= 1 - cosß for this case. Eq. 9 yields fA,B(ß)= sin2ß , and FA,B(ß )= sin2 ß , which is a result true for general .bre processes, see Mecke and Nagel (1980). With Eq. 6 we obtain NA, the mean number of section points per unit area as NA = 1NV ELV . (26) 2 This result is a special case of another well-known stereological formula, namely (11.3.3) in Chiu et al. (2013), where characteristics of germ-grain models are studied. Here the ”grain” is an isotropic line-segment. SEGMENTS OF CONSTANT LENGTH In papers such as Li et al. (1991) and Brandt (1985) line-segment processes and their sections with planes are studied. They derived formulas for the strength of .bre-reinforced materials using calculations for segments of constant length. If in this context the constant length is chosen to be l0 > 0, we obtain with Eq. 8 the marginal distribution function FA,R(r). It holds FV,L(l)= .(l - l0) with the Heaviside function 0, x < 0 .(x)= 1, x . 0 and hr 1 FA,R(r)= 1 - .(l - l0) dll0 0 r , r . l0 l0 = 1, r . l0 for the distribution function of the upper and lower segment length at a typical section point. This distribution is the uniform distribution on [0,l0]. We therefore obtain the mean values of the residual 1 segment lengths ERA = ERA = l0. This result 1 22 coincides with results of Li et al. (1991). Moreover the line-segments in (Li et al., 1991) are considered to be isotropic. Using Eq. 26 we have 1 NA = 2 NV l0. A PARAMETRIC MODEL In Chin et al. (1988), Hegler (1985) and other papers line-segment length and angle distribution functions for segment processes .V with LV independent of BV appear such as -(ml)k FV,L(l)= 1 - e, m,k,l > 0 , the Weibull distribution function, and 1 - exp(-.ß) . FV,B(ß)= , ß .0,,. > 0 . 1 - exp -.. 2 2 These distribution functions are motivated by the process of manufacture of some .bre-reinforced material. They model different effects such as breakage of .bres before moulding or a preferred orientation of .bres after moulding, where the shape parameter . models the orientation density of the line-segments. A large . indicates a major preferential alignment of the line-segments in the z-direction, see also Kacir et al. (1975) and Chin et al. (1988). Note that this parametric model does not include the isotropic case. The Figs. 5 and 6 give an impression of the in.uence of the parameters m, k and . on the probability density functions fV,L and fV,B. Figure 5. In.uence of the parameters k and m on the Weibull probability densitiy function of LV . Figure 6. In.uence of the shape parameter . on the probability density function of polar angle BV . NA Figure 7. The ratio in dependence on shape parameter .. NV Figure 8. The conditional probability density function fA,R1|R2.r2 (r1) in dependence on r2 for Weibull­distributed .bre lengths. This is the probability density function of the length RA of the segment above the 1 plane S, if the segment length RA 2 below S is bounded from above. Here r2 ranges from 0.2 to 1.2 with a step size of 0.2. The distribution of the residual segment lengths can be easily computed using the Eqs. 6–9. For the parameters m = 1 and k = 5 the results are shown in Figs. 7–9. Fig. 7 show that NNVA goes to a limit value as . tends to in.nity. This limit is ELV , the mean length of the line-segments (in the case m = 1 and k = 5 we have the limit value ELV . 0.918). This is plausible since for large . the segments have a preferred orientation in z-direction. Therefore with . › . all line-segments have the polar angle BV = 0 and therefore BA = 0. Applying this case to Eq. 6 we .nd NA = NV ELV . Furthermore, Fig. 8 shows that the conditional distribution of the upper segment length FV,L(r1 + r2) - FV,L(r1) fA,R1|R2.r2 (r1)= (27) r2 1 - FV,L(l) dl 0 is more concentrated if the lower segment length is .xed at small values. Moreover it can be shown that fA,R1|R2.r2 (r1) coincides with fV,L(l) if r2 › 0. Thus with decreasing upper bound of RA the conditional 2 probability density function in Fig. 8 tends to the probability density function of the Weibull distribution with m = 1 and k = 5 in Fig. 8. With increasing r2 the conditional probability fA,R1|R2.r2 (r1) tends to 1 ELV 1 - FV,L(r1) . Since the segment lengths R1 A and RA are independent of the intersection angle BA in 2 this case the conditional probability density function fA,R1|R2.r2 is independent of .. Image Anal Stereol 2014;33:55-64 Figure 9. The probability density function fA,B(ß) of intersection angles in dependence on orientation distribution parameter . which varies from 0 to 3. In Fig. 9 we see that with increasing . the segments at the typical section point show a preferred direction in the z-axis. Since the segment lines tend to have a small polar angle BV for large ., the segments intersecting S have a small polar angle BA too. For . › 0 the probability density function fA,B(ß) tends to cosß. Note that this is not the isotropic case since for isotropic line-segments it holds fA,B(ß)= sin2ß . MAXIMUM AND MINIMUM RESIDUAL SEGMENT LENGTH We investigate the stochastic variables M = max{RA 1 ,R2 A} and m = min{RA 1 ,R2 A}, which are only in special cases stochastic independent. We therefore consider .rst the joint distribution function FA,m,M(rm,rM) with rm,rM > 0. Furthermore we consider the marginal distribution functions FA,M(r)= P(M . r)= P(RA 1 . r,RA 2 . r) and FA,m(r)= P(m . r)= 1 - P(RA 1 > r,RA 2 > r) for r > 0. Using simple ideas of probability we obtain . . .. . . FA,R1,R2 (rm,rM) +FA,R1,R2 (rM,rm) rm . rM and FA,m(r)= 2FA,R(r) - FA,R1,R2 (r,r) . With Eqs. 7 and 8 we can relate these distribution functions with the distribution functions of .V : FA,m,M(rm,rM)= hrm 1 2FV,L(l + rM) - FV,L(l + rm) - FV,L(l) dl , ELV 0 (28) FA,M(r) = 1 ELV hr FV,L(l + r) - FV,L(l) dl , (29) 0 FA,m(r) = 1 ELV hr 2 - FV,L(l) - FV,L(l + r) dl . (30) 0 The marginal distributions simplify with FV,L(l)= 1 - FV,L(l): hr 1 FA,M(r)= FV,L(l) - FV,L(l + r) dl ELV 0 and hr 1 FA,m(r)= FV,L(l)+ FV,L(l + r) dl . ELV 0 In case of Weibull-distributed lengths it holds FV,L(l)= 1 - e-(ml)k and FV,L(l)= e-(ml)k . The corresponding results are sketched in Fig. 10. FA,m,M(rm,rM)= -FA,R1,R2 (rm,rm) FA,R1,R2 (rM,rM) rM . rm maximum residual length, FA,m(l) and FA,M(l), for FA,M(r)= FA,R1,R2 (r,r) Weibull-distributed total lengths with m = 1 and k = 5. ACKNOWLEDGEMENTS The authors thank K.G. van den Boogaart for inspiring discussion during a seminar at the Institute of Stochastics of the TU Freiberg and U. Hampel, M. Bieberle and S. 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