The chord length distribution function of a non-convex hexagon

Communications in Applied and Industrial Mathematics ISSN 2038-0909

Research article Commun. Appl. Ind. Math. 9 (1), 2018, 20–34

DOI: 10.1515/caim-2018-0002

The chord length distribution function of a non-convex hexagon Uwe B¨ asel1* , Vittoria Bonanzinga2 , Andrei Duma3 1

HTWK Leipzig, University of Applied Sciences, Faculty of Mechanical and Energy Engineering, Germany 2

Universit` a degli Studi di Reggio Calabria, Dipartimento di Ingegneria dell’Informazione, delle Infrastrutture e dell’Energia Sostenibile, Italy 3 FernUniversit¨ at in Hagen, Fakult¨at f¨ ur Mathematik und Informatik, Germany *

Email address for correspondence: [email protected] Communicated by Mario Primicerio Received on 12 19, 2016. Accepted on 12 20, 2017.

Abstract In this paper we obtain the chord length distribution function of a non-convex equilateral hexagon and then derive the associated density function. Finally, we calculate the expected value of the chord length. Keywords: Non-convex hexagon, chord length distribution function, multiple chord distribution, chord power integrals AMS subject classification: 60D05, 52A22

1. Introduction On the one hand, chord length distributions of plane figures and spatial bodies are studied for theoretical reasons (see [17], [12], [7]), on the other hand, there are numerous practical applications such as in acoustics, ecology, image analysis, stereology, and reactor design (see [14, p. 6195] for further bibliographical references). An important application one finds in the small-angle scattering (SAS) for the investigation of material structures using neutron beams or X-rays in order to get information about the average size and the size distribution of particles [8], [9]. The chord length distribution functions for a number of planar convex figures are already known. Please refer to the following results: • • • •

regular hexagon [4], arbitrary triangles [5], [3], rectangular trapezium [6], isosceles trapezium [16],

c 2018 Uwe Basel,Vittoria

¨ Bonanzinga,Andrei Duma, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

The chord length distribution function of a non-convex hexagon

• every regular polygon [11], [2], where in [2] also the distribution function is determined for the distance between two random points uniformly distributed in a regular polygon. There are straight lines that cross a non-convex body more than one time. So for non-convex bodies we have to distinguish between the chord length distribution function of the one chord distribution (OCD-function) and those of the multiple chord distribution (MCD-function) [14, p. 6195]. If the intersection S = G ∩ F of a random straight line G and a non-convex body F consists of several line segments Si , i = 1, 2, . . . , n, every Si has to be counted as one chord of length |Si | in order to obtain the MCD-function. In the case of the OCD-function, all line segments of S are considered as P one chord of length |S| = ni=1 |Si |. The OCD-function and the MCD-function for a non-convex polygon (Kshaped pentagon) have been determined in [1], the MCD-function by means of simulation and analytical computation, the OCD-function by means of simulation. In the present paper the notion chord length distribution function always means MCD-function. We determine this function and the associated density function for the non-convex equilateral hexagon H (see Fig. 1) in Section 2, and draw some further conclusions in Section 3.

Figure 1.

The non-convex equilateral hexagon H (grey-shaded)

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¨ U. Basel, V. Bonanzinga, A. Duma

2. The distribution function The measure m for the set S of all chords of H is defined by Z 2π Z Z #(Si : Si ⊂ G) dp dϕ #(Si : Si ⊂ G) dG = m(S) = G ∩ H 6= ∅

0

R+

(see [1, p. 142]), where dG = dp dϕ is the density for sets of straight lines, ϕ is the angle of the normal of the line G with the x axis, and p is the distance of G from the origin. So the equation of G is given by x cos ϕ + y sin ϕ − p = 0 . Up to a constant factor, dG is the only density that is invariant under planar motions [15, p. 28]. Considering p as the signed distance from the origin, we may restrict ϕ to the interval 0 ≤ ϕ ≤ π and get Z πZ m(S) = #(Si : Si ⊂ G) dp dϕ . 0

R

For the calculation of m(S) we need the width w(ϕ) of H in the direction ϕ. This width function is, as easily verifiable, given by  w1 (ϕ) , 0 ≤ ϕ < π/6 ,     w (ϕ) , π/6 ≤ ϕ < π/2 2 w(ϕ) =  −w2 (−ϕ) , π/2 ≤ ϕ < 5π/6 ,    −w1 (−ϕ) , 5π/6 ≤ ϕ ≤ π , where w1 (ϕ) =

√

3 r cos

π 6

−ϕ



and w2 (ϕ) = 2r cos

π 3

 −ϕ .

We note that w(ϕ) is symmetrical with respect to the line ϕ = π/2. For 0 ≤ ϕ ≤ π/6 the lines in the strip of the width π  t2 (ϕ) = r sin −ϕ 6 (see Fig. 4) have to be counted twice, since each of these lines carries two chords. Taking into account the symmetry of H, we get Z π/2 Z m(S) = 2 #(Si : Si ⊂ G) dp dϕ 0

R

(Z =2

π/2

Z w(ϕ) dϕ +

0

) t2 (ϕ) dϕ

0

22

π/6

The chord length distribution function of a non-convex hexagon

( =2

√

3 2r + r 2

√

! +

3 r− r 2

!) = 6r .

Alternatively, this result immediately follows from the measure for the set of all lines that intersect a regular hexagon. For every bounded convex set, this measure is equal to its perimeter (see e. g. [15, p. 30]), hence for the regular hexagon of side length r, it is equal to 6r. The difference between the measures for the sets of lines intersecting the regular hexagon and the non-convex hexagon H is equal to Z

π/6

t2 (ϕ) dϕ .

2 0

But this is also the measure for the set of all lines G intersecting H with #(Si : Si ⊂ G) = 2. The measure for all chords S 0 ∈ S of length |S 0 | ≤ s is given by Z 0 0 #(Si : Si ⊂ G, |Si | ≤ s) dG m(S : |S | ≤ s) = G ∩ H 6= ∅ πZ

Z

#(Si : Si ⊂ G, |Si | ≤ s) dp dϕ .

= 0

R

It follows that the distribution function F for the length of a random chord S 0 of H is defined by F (s) = P (|S 0 | ≤ s) =

m(S 0 : |S 0 | ≤ s) m(S)

(see [1], Eq. (2)). In the present case, therefore F (s) =

1 m(S 0 : |S 0 | ≤ s) . 6r

Concerning the chord length s, it is necessary to distinguish between the four cases: √ √ √ √ 3 3 1) 0 ≤ s ≤ r , 2) r ≤ s ≤ r , 3) r ≤ s ≤ 3 r , 4) 3 r ≤ s ≤ 2r . 2 2 In the following, the values √ √ 3r 3r ϕ0 (s) = arccos , ϕ1 (s) = arcsin , 2s 2s

√ ϕ2 (s) = arcsin

of the angle ϕ are needed for the integration limits.

23

3r s

¨ U. Basel, V. Bonanzinga, A. Duma

Now we consider Case 1: a) For fixed value of the angle ϕ ∈ [0, π/6], the set of chords with length ≤ s consists of three strips (see Fig. 2) with the widths       s 1 s 1 π π h1 (s, ϕ) = √ , h2 (s, ϕ) = √ , + sin 2ϕ + − sin 2ϕ − 6 6 3 2 3 2   s 1 h3 (s, ϕ) = √ cos 2ϕ − . 2 3 We note that the first two strips partially overlap if the respective lines carry two chords. b) For ϕ ∈ [π/6, π/2] the set of chords with length ≤ s consists of two non-overlapping strips (Fig. 3) of widths h1 (s, ϕ) (see a) and     s 2π 1 h4 (s, ϕ) = √ cos 2ϕ − − . 3 2 3

Figure 2.

Case 1a

Figure 3.

Case 1b

Therefore, the restriction F1 (s) of the chord length distribution function F (s) for Case 1 is given by (Z π/6 1 F1 (s) = [h1 (s, ϕ) + h2 (s, ϕ) + h3 (s, ϕ)] dϕ 3r 0 ) √ Z π/2 15 3 + π s √ . + [h1 (s, ϕ) + h4 (s, ϕ)] dϕ = 36 3 r π/6

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The chord length distribution function of a non-convex hexagon

Now we consider Case 2: a) ϕ ∈ [0, ϕ0 (s)] (Fig. 4): The set of chords with length ≤ s consists of three strips of widths  π  π t1 (ϕ) = r sin ϕ + , t2 (ϕ) = r sin −ϕ , 6 6 and (see Case 1a) h3 (s, ϕ). b) ϕ ∈ [ϕ0 (s), π/6]: The set of chords with length ≤ s consists of three strips of widths h1 (s, ϕ), h2 (s, ϕ) and h3 (s, ϕ) (see Case 1a). c) ϕ ∈ [π/6, ϕ1 (s) − π/6]: There are two strips of widths h1 (s, ϕ) and h4 (s, ϕ) (see Case 1b). d) ϕ ∈ [ϕ1 (s) − π/6, 5π/6 − ϕ1 (s)] (Fig. 5): One gets   s 1 − cos 2ϕ , `2 (ϕ) = r cos ϕ `1 (s, ϕ) = √ 3 2 and (see Case 1b) h4 (s, ϕ). e) ϕ ∈ [5π/6 − ϕ1 (s), π/2]: Here we have two strips of widths h1 (s, ϕ) and h4 (s, ϕ) (see Case 1b).

Figure 4.

Case 2a

Figure 5.

Case 2d

Hence the restriction F2 of F for Case 2 is given by (Z ϕ0 (s) 1 F2 (s) = [t1 (ϕ) + t2 (ϕ) + h3 (s, ϕ)] dϕ 3r 0 Z π/6 + [h1 (s, ϕ) + h2 (s, ϕ) + h3 (s, ϕ)] dϕ ϕ0 (s)

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¨ U. Basel, V. Bonanzinga, A. Duma

Z

ϕ1 (s)−π/6

+

[h1 (s, ϕ) + h4 (s, ϕ)] dϕ π/6

Z

5π/6−ϕ1 (s)

+

[`1 (s, ϕ) + `2 (ϕ) + h4 (s, ϕ)] dϕ ϕ1 (s)−π/6

Z

)

π/2

[h1 (s, ϕ) + h4 (s, ϕ)] dϕ

+ 5π/6−ϕ1 (s)

r √  r 2 15 3 + π − 12 ϕ0 (s) s 1 √ = + 4−3 . r 4 s 36 3 We consider Case 3: a) ϕ ∈ [0, ϕ1 (s) − π/6]: t1 (ϕ), t2 (ϕ) and h3 (s, ϕ) (see Case 2a). b) ϕ ∈ [ϕ1 (s) − π/6, π/6] (Fig. 6): t1 (ϕ), t2 (ϕ) (see Case 2a) and u1 (ϕ) = r sin

π 6

 −ϕ ,

   s 1 π u2 (s, ϕ) = √ + sin 2ϕ + − r cos ϕ . 6 3 2

c) ϕ ∈ [π/6, π/2 − ϕ1 (s)] (Fig. 7): `1 (s, ϕ), `2 (ϕ) (Case 2d) and u2 (s, ϕ) (Case 3b).

Figure 6.

Case 3b

Figure 7.

Case 3c

d) ϕ ∈ [π/2 − ϕ1 (s), π/6 + ϕ1 (s)]: `1 (s, ϕ), `2 (ϕ) and h4 (s, ϕ) (see Case 2d). e) ϕ ∈ [π/6+ϕ1 (s), π/2] (Fig. 8): The lengths of all chords in the direction perpendicular to ϕ are ≤ s, and there are no lines that carry two chords. Hence the width of the strip is equal to the width function w2 (ϕ).

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The chord length distribution function of a non-convex hexagon

Figure 8.

Case 3e

Figure 9.

Case 4a

So we have 1 F3 (s) = 3r

(Z

ϕ1 (s)−π/6

[t1 (ϕ) + t2 (ϕ) + h3 (s, ϕ)] dϕ 0

Z

π/6

+

[t1 (ϕ) + t2 (ϕ) + u1 (ϕ) + u2 (s, ϕ)] dϕ ϕ1 (s)−π/6

Z

π/2−ϕ1 (s)

+

[`1 (s, ϕ) + `2 (ϕ) + u2 (s, ϕ)] dϕ π/6

Z

π/6+ϕ1 (s)

+

Z

π/2

[`1 (s, ϕ) + `2 (ϕ) + h4 (s, ϕ)] dϕ + π/2−ϕ1 (s)

) w2 (ϕ) dϕ

π/6+ϕ1 (s)

r √  r 2 2 3 3 − 7π + 24 ϕ1 (s) s 1 √ = − + 4−3 . 3 r 12 s 36 3 Case 4: a) ϕ ∈ [0, π/2 − ϕ2 (s)] (Fig. 9): The lengths of all chords in the direction perpendicular to ϕ are ≤ s. The lines in the strip of width of t2 (ϕ) carry two chords. Therefore, w1 (ϕ) + t2 (ϕ) is the measure for all chords with length ≤ s for fixed angle ϕ. b) ϕ ∈ [π/2 − ϕ2 (s), π/6]: t1 (ϕ), t2 (ϕ), u1 (ϕ), u2 (s, ϕ) (see Case 3b). c) ϕ ∈ [π/6, ϕ2 (s) − π/6]: `1 (s, ϕ), `2 (ϕ), u2 (s, ϕ) (see Case 3c). d) ϕ ∈ [ϕ2 (s) − π/6, π/2]: w2 (ϕ) (see Case 3e).

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¨ U. Basel, V. Bonanzinga, A. Duma

We get 1 F4 (s) = 3r

(Z

π/2−ϕ2 (s)

[w1 (ϕ) + t2 (ϕ)] dϕ 0

Z

π/6

[t1 (ϕ) + t2 (ϕ) + u1 (ϕ) + u2 (s, ϕ)] dϕ

+ π/2−ϕ2 (s)

Z

ϕ2 (s)−π/6

+

Z

)

π/2

[`1 (ϕ) + `2 (ϕ) + u2 (s, ϕ)] dϕ + π/6

w2 (ϕ) dϕ ϕ2 (s)−π/6

2 π − 3 ϕ2 (s) s 2 √ = − + 3 r 3 6 3

r 1−3

 r 2 s

.

By simplifying the previous results, one finds the following theorem. Theorem 2.1. The distribution function F of the chord non-convex hexagon H is given by  0 if        s π    if 15 + √   36r 3        5s π s ψ(h/s) R(h/s)   if   3 − √3 36r + 3 √3 r + 2   F (s) =  7π 2s ψ(h/s) R(h/s) 2 s   √ − 3− √ − if +   3 6 3 36r 3 3r      2 πs s ψ(2h/s) 2R(2h/s)   √ − √ + + if    3 6 3r 3 2 3r     1 if where

length s of the −∞ < s < 0 , 0 ≤ s < h, h ≤ s < r, r ≤ s < 2h , 2h ≤ s < 2r , 2r ≤ s < ∞ ,

√ h=

3r , 2

ψ(x) = arcsin x ,

R(x) =

p

1 − x2 .

The graph of F is shown in Fig. 10 (solid line). As a comparison we show the distribution function of the regular hexagon of side length r (dashed line) according to [10] and [2, Theorem 1]. With 1 dψ(x) =√ , dx 1 − x2 x = χ(s) =

dR(x) x = −√ , dx 1 − x2

a , s

dχ(s) a =− 2, ds s

28

The chord length distribution function of a non-convex hexagon

we easily get the density function f of the chord length s:

f (s) =

 0 if −∞ < s < 0 ,       f1 (s) if 0 ≤ s < h ,      f2 (s) if h ≤ s < r ,  f3 (s) if r ≤ s < 2h ,         f4 (s) if 2h ≤ s < 2r ,   0 if 2r ≤ s < ∞ ,

where

 1 π √ , 15 + 36r 3     π 5 1 h h2 3− √ + √ ψ(h/s) − + 3 , s R(h/s) 2s R(h/s) 3 36r 3 3 r     1 2 h h2 7π − √ ψ(h/s) − + 3 , − 3− √ s R(h/s) 6s R(h/s) 3 36r 3 3 r   π 1 2h 8h2 − √ + √ ψ(2h/s) − + 3 . s R(2h/s) 3s R(2h/s) 6 3r 2 3r

 f1 (s) = f2 (s) = f3 (s) = f4 (s) =

For the right-hand limits √ at the two discontinuities of f (see Fig. 11), with the abbreviation h = 3 r/2 we find

lim

s→h+0

f (s) = lim f2 (s) = ∞ , s→h

lim

s → 2h + 0

f (s) = lim f4 (s) = ∞ . s → 2h

Furthermore, we have √ 6 + 3π 0.317817 lim f (s) = lim f3 (s) = ≈ . s → 2h − 0 s → 2h − 0 36 r r

29

¨ U. Basel, V. Bonanzinga, A. Duma

FHsL 1.0

0.8

0.6

0.4

0.2

sr 3 2

1

3

2

Figure 10. Chord length distribution functions of H (solid line) and of the regular hexagon of side length r (dashed line)

r ‰ f HsL 1.0 0.8 0.6 0.4 0.2

sr 3 2

1

3

2

Figure 11. Chord length density function of H (solid line); chord length density function of the regular hexagon of side length r (dashed line) according to [10]

30

The chord length distribution function of a non-convex hexagon

3. Conclusions The linear part f1 (s) of the density function f (s) in Case 1 can also be obtained by Gates’ formula [7, p. 866-867] " # n X 1 n+ f1 (s) = (π − γi ) cot γi 2L i=1

for convex polygons, where L is the perimeter of the polygon and n is the number of its internal angles γi . Since H is non-convex, we only consider the internal angles γi ≤ π/2. With γ1 = γ2 =

π , 3

γ3 = γ4 = γ5 =

2π 3

we get " #   5 X 1 1 π f1 (s) = 5+ . (π − γi ) cot γi = 15 + √ 12r 3 36r i=1 For the expected value E[s] of the chord length s we find Z

∞

E[s] =

2r

Z s dF (s) =

−∞

Z =

sf (s) ds 0

h

Z

r

sf1 (s) ds + 0



πr 5r √ + = 96 3 32 πr = √ . 2 3

2h

Z sf2 (s) ds +

h



 +

Z

r

29πr r √ + 288 3 96

2r

sf3 (s) ds +

sf4 (s) ds 2h



 +

13πr r √ + 72 3 12



 +

5πr r √ − 24 3 4



√ The area A of H is equal to 3 r2 , and the length L of the boundary ∂H is equal to 6r, hence √ √ πA 3 πr2 3 πr πr = = = √ = E[s] . L 6r 2·3 2 3 We denote by In , n = 0, 1, 2, . . . , the n-th chord power integral defined by Z In = sn dG [15, p. 46, Eq. (4.3)] . G ∩ K 6= ∅

31

¨ U. Basel, V. Bonanzinga, A. Duma

In [15, p. 47, Eq. (4.7)] (see also [13, p. 94]) one finds that the expected value E[s] of a bounded convex set K is given by E[s] =

I1 I0

with I0 = L

and I1 = πA .

This result for convex sets is also true for the non-convex hexagon H. We note, however, that the measure of all lines intersecting H, Z

Z dG = 2

G ∩ H 6= ∅

π/2

w(ϕ) dϕ = 4r +

√

3r,

0

is equal to the perimeter of the convex hull of H, and thus is not equal to the length L of ∂H. In [14, p. 6201-6203] is shown that E[s] = πA/L applies also in general for two planar convex overlapping sets, and, clearly, we may consider H as the union of two overlapping convex sets. Our results may be applied in the different fields mentioned in the introduction of the present paper, especially in the small-angle scattering (SAS). It is possible to use H in order to disprove conjectures concerning chord length distributions of non convex sets, or check that such a conjecture might be true. Today, there is much more known about distributions for convex sets than for non convex sets. Furthermore, the results might be used as base for the calculation of the distribution of the distance between two points chosen uniformly at random in the non-convex set H. As already mentioned, for regular polygons this step was done in [2]. Such distributions serve as simple models for the study of wireless ad hoc networks in plane areas (cells). REFERENCES 1. Uwe B¨ asel, Andrei Duma: Verteilungsfunktionen der Sehnenl¨ ange eines nichtkonvexen Polygons, Fernuniversit¨at Hagen: Seminarberichte aus der Fakult¨at f¨ ur Mathematik und Informatik, 84 (2011), 141-151. https://www.fernuni-hagen.de/imperia/md/content/ fakultaetfuermathematikundinformatik/forschung/ berichte mathematik/berichte band 84.pdf 2. Uwe B¨ asel: Random chords and point distances in regular polygons, Acta Math. Univ. Comenianae, 83, 1 (2014), 1-18. http://www.emis. de/journals/AMUC/ vol-83/ no 1/ baesel/baeselrea.pdf 3. Salvino Ciccariello: The isotropic correlation function of plane figures: the triangle case, Journal of Physics: Conference Series 247

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The chord length distribution function of a non-convex hexagon

(2010) 012014, XIV International Conference on Small-Angle Scattering (SAS09), 1-10. http://iopscience.iop.org/article/10.1088/1742-6596/ 247/1/012014/pdf 4. Vincenzo Conserva, Andrei Duma: Schnitte eine regul¨ aren Hexagons im Buffon- und Laplace-Gitter, Fernuniversit¨at Hagen: Seminarberichte aus der Fakult¨ at f¨ ur Mathematik und Informatik, 78 (2007), 29-36. 5. Andrei Duma, Sebastiano Rizzo: Chord length distribution functions for an arbitrary triangle, Rend. Circ. Mat. Palermo, Serie II, Suppl. 81 (2009), 141-157. http://math.unipa.it/∼circmat/Suppl%2081%20PDF. pdf 6. Andrei Duma, Sebastiano Rizzo: La funzione di distribuzione di una corda in un trapezio rettangolo, Rend. Circ. Mat. Palermo, Serie II, Suppl. 83 (2011), 147-160. http://math.unipa.it/∼circmat/ Supplemento 83.pdf 7. John Gates: Some properties of chord length distributions, J. Appl. Prob., 24 (1987), 863-874. https://www.jstor.org/stable/3214211 8. Wilfried Gille: Chord length distributions and small-angle scattering, Eur. Phys. J. B, 17 (2000), 371-383. https://link.springer.com/content/ pdf/10.1007%2Fs100510070116.pdf 9. Wilfried Gille: Particle and Particle Systems Characterization: Small-Angle Scattering (SAS) Applications, CRC Press, Boca Raton/London/New York, 2014. 10. H. S. Harutyunyan: Chord length distribution function for regular hexagon, Uchenie Zapiski, Yerevan State University, 1 (2007), 17-24. http://www.ysu.am/files/2.CHORD%20LENGTH% 20DISTRIBUTION%20FUNCTION.pdf 11. H. S. Harutyunyan, V. K. Ohanyan: The chord length distribution function for regular polygons, Adv. Appl. Prob. (SGSA), 41 (2009), 358-366. https://www.jstor.org/stable/27793882 12. C. L. Mallows, J. M. C. Clark: Linear-intercept distributions do not characterize plane sets, J. Appl. Prob., 7 (1970), 240-244. 13. Arakaparampil M. Mathai: An Introduction to Geometrical Probability, Gordon and Breach, Australia, 1999. 14. Alain Mazzolo, Benoˆıt Roesslinger, Wilfried Gille: Properties of chord length distributions of nonconvex bodies, Journal of Mathematical Physics, 44, 12 (2003), 6195-6208. http://aip.scitation.org/doi/abs/10. 1063/1.1622446

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15. Luis A. Santal´ o: Integral Geometry and Geometric Probability, AddisonWesley, London, 1976. 16. Loredana Sorrenti: Chord length distribution functions for an isosceles trapezium, General Mathematics, 20, 1 (2012), 9-24. http://depmath. ulbsibiu.ro/genmath/gm/vol20nr1/02 Sorrenti/02 Sorrenti.pdf 17. Rolf Sulanke: Die Verteilung der Sehnenl¨ angen an ebenen und r¨ aumlichen Figuren, Math. Nachr., 23 (1961), 51-74. http:// onlinelibrary.wiley.com/doi/10.1002/mana.19610230104/epdf

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