Olivier Devillers
email: olivier.devillers@inria.fr

Sylvain Lazard
email: sylvain.lazard@inria.fr

Drawing K n in Three Dimensions with One Bend per Edge, revisited *

We present a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossing-free with at most one bend per edge; the bend points are also placed at integer grid points. The drawing is determined by an incremental algorithm and the observed behavior of the algorithm produces an output of volume close to quadratic.

Introduction

Although graphs are usually drawn in two dimensions, several authors have considered 3D-drawings. It is clear that if we place vertices in general positions and draw edges as line segments, then the edges are disjoint and crossing free drawings in this context is a trivial problem. The problem becomes interesting if vertices are constrained to the integer grid and we try to minimize the volume of the axis aligned bounding box of the drawing. 1Of course the 3D-drawing problem arises for any kind of graph and several classes of graphs has been considered. Such classes are defined by parameters or properties such as outerplanarity [START_REF] Felsner | Straight-line drawings on restricted integer grids in two and three dimensions[END_REF], planarity [START_REF] Di Battista | On the queue number of planar graphs[END_REF][START_REF] Calamoneri | 3D straight-line grid drawing of 4-colorable graphs[END_REF], colorability [START_REF] Calamoneri | 3D straight-line grid drawing of 4-colorable graphs[END_REF][START_REF] Pach | Three-dimensional grid drawings of graphs[END_REF], stack-number, queuenumber, track-number [START_REF] Dujmović | Stacks, queues and tracks: Layouts of graph subdivisions[END_REF], or prescribed position for vertices [START_REF] Dujmović | On point-sets that support planar graphs[END_REF][START_REF] Everett | Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices[END_REF][START_REF] Meijer | Point-set embedding in three dimensions[END_REF][START_REF] Wood | Three dimensional graph drawing with fixed vertices and one bend per edge[END_REF], In this paper we focus on the case of the complete graph K n .

If the edges are constrained to be straight line segments, Cohen, Eades, Lin and Ruskey [START_REF] Cohen | Three-dimensional graph drawing[END_REF] proposed a crossing free drawing of the complete graph K n within an optimal bounding box of volume Θ(n 3 ). Some papers consider the possibility of allowing some bends per edge. Bose, Czyzowicz, Morin, and Wood [START_REF] Bose | The maximum number of edges in a three-dimensional grid-drawing[END_REF] showed that the number of edges is a lower bound on the volume of the bounding box, even in the presence of bends, establishing a quadratic lower bound on the volume of the complete graph. Dyck, Joevenazzo, Nickle, Wilsdon and Wismath [START_REF] Dyck | Drawing K n in Three Dimensions with Two Bends Per Edge[END_REF] achieved such a quadratic construction with two bends per edge. If we restrict the number of bends to at most one per edge, Morin and Wood [START_REF] Morin | Three-dimensional 1-bend graph drawings[END_REF] presented a one-bend drawing of K n with a volume

X Z Y Z Figure 1: A collinear drawing of K 8 . of O( n 3 log 2 n
), later improved by Devillers, Everett, Lazard, Pentcheva, and Wismath to O(n 2.5 ) [START_REF] Devillers | Drawing Kn in Three Dimensions with One Bend per Edge[END_REF].

In this paper, we analyze the techniques that yielded these results for the one-bend drawings and their limitation, and we propose an algorithm that bypass these limitations and provides good experimental results. The actual computation of K n for n up to 100 shows a behavior close to quadratic and we conjecture that the actual volume of our construction is O(n 2 log 2 n).

Preliminaries

Collinear drawing of the complete graph

The first result proposed by Morin and Wood is about a collinear drawing of the graph, that is a drawing where all vertices lie on a line.

Lemma 1 ([12, Lemmas 3 & 4 ]). A one-bend collinear drawing of K n with vertices on the Y -axis can be achieved within a box of size

N X × n × O( n 2 N X ) ∀N X ∈ [1, n 2 ] of volume Θ(n 3 ).
Sketch. Refer to Figure 1. We assume that the vertices are placed at coordinates (0, i, 0) for 0 ⩽ i < N Y with N Y > n. Then in projection in the XZ plane, the two pieces of an edge look like a single line segment between the origin and a grid point: the projection of the bend.

-Realisability. A 2D grid point is said visible from the origin if its two cartesian coordinates are relatively prime. One can pick K grid points visible from the origin

X Z Y Z X Y Z Figure 2: Drawing of K 5,4
. Each edge has two segments, one from the vertex in the first set to its bend in blue, another from the bend to the vertex in the second set in red. Lemma 2]. Going to three dimensions, these grid points define "pages" around the Y -axis. Then, for each edge of the complete graph, we pick the bend in a separate page. There are n(n-1) 2 such edges, thus the graph can be drawn in a box of size

in the XZ plane with x ∈ [0, N X ] and z ∈ [0, N Z ] provided that N X N Z ⩾ 2π 2 3 K [12,
N X × N Y × N Z of volume not bigger than π 2 3 n 3 if N Y = n.
-Lower bound.Several edges can be drawn on the same page, but nevertheless a quadratic number of bends is necessary. Consider two edges from (0, i, 0) to (0, j, 0), and from (0, k, 0) to (0, ℓ, 0) such that 0 ⩽ i, k < n 2 < j, ℓ << n. These two edges must have bends with different projections in XZ-plane. The lower bound follows since the number of such edges is quadratic.

Bi-collinear drawing of the complete bi-partite graph

The construction proposed by Morin and Wood [12, §3] uses, as building blocks, drawings of complete bi-partite graphs with exactly one bend point per edge. These drawings are called one-bend bi-collinear drawings because the two sets of vertices of K n,m are furthermore constrained to lie on two different lines parallel to the Y -axis.

Lemma 2 ([12]

). A one-bend bi-collinear drawing of K n,m can be achieved within Θ(nm) volume.

Sketch. The lower bound is trivial, since K n,m has nm edges and thus must have nm bends [START_REF] Bose | The maximum number of edges in a three-dimensional grid-drawing[END_REF]. Now refer to Figure 2. To realize the graph, we place n points at coordinates (0, i, 0) for 0 ⩽ i < n and m points at coordinates (x, y + j, z) for 0 ⩽ j < m and (x, y, z) ∈ Z 3 . Then, the bend point for the edge between (0, i, 0) and (x, y + j, z) is placed at (x ′ , j, z ′ + i) 0 < x ′ < x and z ′ ∈ Z.

By choosing suitable values, e.g., x ′ = 1, z ′ = 0, (x, y, z) = (2, 0, 0), the volume of this drawing is 3nm. Notice that this construction is contained in two convex volumes, that we call wedges, which are the convex hulls of, respectively, the bends and the n vertices, and the bends and the m vertices.

Disjointness of pieces of edges is granted by considering the different planes parallel to the Y -axis they belong to (see Figure 2-right).

Remark 3. Notice that this construction still works if we place the bends of the edges from (0, i, 0) to (x, y + j, z) at (x ′ , y ij , z ′ + i) (instead of (x ′ , j, z ′ + i)) where (y i,j ) j are sequences of integers that are strictly increasing with j (sequences may depend on i).

Previous constructions

The idea of the Morin-Wood and the Devillers-Everett-Lazard-Pentcheva-Wismath constructions is to cut the set of n vertices in k groups of size n k . To place the groups on some lines parallel to the Y -axis and draw for each group a drawing of its complete graph in a box using Lemma 1 and for each pair of groups a drawing of its complete bi-partite graph in two wedges using Lemma 2. These drawings are realized such that all the boxes and all the wedges are pairwise interior disjoints. The complete graphs are drawn below the vertices, the bi-partite graphs are drawn above and the number of groups is tuned to balance the heights of the constructions above and below.

The Morin-Wood construction.

In the Morin-Wood construction [START_REF] Morin | Three-dimensional 1-bend graph drawings[END_REF], the vertices are split in a logarithmic number of groups. The groups are placed in the XY plane such that (i) all vertices have different Y -coordinates, and (ii) vertices of the same group are at consecutive Ycoordinates and at the same (even) X-coordinate. The boxes for the complete graph of each group are below the XY plane, and the wedges are above it. They show that this construction is possible with a height of O(

n 2 log 3 n ) yielding a box of size [0, log 4 n] × [0, n] × [-O( n 2 log 3 n ), O( n 2 log 3 n )] of volume O( n 3 log 2 n ).

2.3.2

The Devillers-Everett-Lazard-Pentcheva-Wismath construction.

In this construction [START_REF] Devillers | Drawing Kn in Three Dimensions with One Bend per Edge[END_REF], the number of groups is k = n 1 4 . As in the previous construction the groups are placed with properties (i) and (ii) but at different Z-coordinates. Placing the groups of vertices on a parabola in the XZ plane allows to draw the bipartite graphs using less height than in the previous construction. Namely the height is O(n 5 4 ) giving a box of size [0, 2n

1 4 ] × [0, n] × [-O(n 5 4 ), O(n 5 4 )] of volume of O(n 5
2 ). Figure 3 presents such a drawing.

New construction: mixing the pencils

Requiring that the different wedges and boxes used in the construction are pairwise disjoint impose strong restrictions on the possible constructions using this paradigm. It is possible to imagine situations where some wedges intersect although the different line segments in the wedges remain crossing free.

We thus propose the following strategy (refer to Figure 4): -We split the vertices in k = √ n groups of k vertices (without loss of assumption, we assume n to be the square of an integer).

-Vertex v i,j , 0 ⩽ i, j < k is placed at coordinates (2i, j, 0) in XY -plane.

-The complete graph of group v i,⋆ is drawn in the box [2i, 2i+1]×[0, k]×[-k 2 , 0] using Lemma 1. Notice that these boxes are pairwise disjoint.

-The bend for the edge from v i,j to v i ′ ,j ′ with i < i ′ is placed at e i,j,i ′ ,j ′ = (i ′ + i, y i,j,i ′ ,j ′ , z i ′ -i k -j) where z 1 = 1, z i = i i-1 z i-1 + 1, and y i,j,i ′ ,j ′ will be determined algorithmically later (before Theorem 7).

Figure 4 depicts this construction for k = 5. Before studying the disjointness of the edges, we start by a preliminary lemma on the heights of the bend points.

Lemma 4.

∀i ⩾ 3, 1 + i ln i < z i < 2i ln i.

Proof. First, this is true for z 3 = 6 < 2 • 3 ln 3 ≃ 6.59 and z 3 > 1 + 3 ln 3 ≃ 4.3. By induction, we have:

zi = i i -1 zi-1 + 1 i i -1 zi-1 + 1 ⩽ zi ⩽ i i -1 zi-1 + 2 i i -1 (1 + (i -1) ln(i -1)) + 1 < zi < i i -1 2(i -1) ln(i -1) + 2 by induction hypothesis 1 + i ln i < i 1 i -1 + ln(i -1) + 1 < zi < 2i ln(i -1) + 1 i < 2i ln i.
where we use for the last leftmost and rightmost inequalities the fact that

1 i < i i-1 dx x < 1 i-1
and thus that ln i = ln(i -1) The edge from v i,j to v i ′ ,j ′ consists of an upgoing line segment between v i,j and e i,j,i ′ ,j ′ and a downgoing line segment between e i,j,i ′ ,j ′ and v i ′ ,j ′ . We first show the following lemma: Lemma 5. Two edges starting (resp. finishing) within the same group of vertices are interior disjoint.

+ i i-1 dx x ∈ (ln(i -1) + 1 i , ln(i -1) + 1 i-1 ). toto X Z i 1 = i 2 e i1,⋆,i ′ 1 ,⋆ i ′ 1 = i ′ 2 e i2,⋆,i ′ 2 ,⋆ Cases (i-ii) toto X Z i 1 = i 2 i ′ 2 = i ′ 1 + 1 e i2,⋆,i ′ 2 ,⋆ i ′ 1 Case (iii) e i1,⋆,i ′ 1 ,⋆
Proof. As in previous construction, one can compare the wedges in XZ projection. Refer to Figure 5. Consider two segments: [v i1,j1 ; e i1,j1,i ′

1 ,j ′ 1 ] and [v i2,j2 ; e i2,j2,i ′ 2 ,j ′ 2 ]
and assume without loss of generality that i 1 = i 2 and i ′ 1 ⩽ i ′ 2 . We distinguish several cases:

(i) If i ′ 1 = i ′ 2 and j 1 = j 2 , then the two segments coincide in XZ projection and intersect only at their origin in 3D (if the two edges are distincts).

(ii) If i ′ 1 = i ′ 2 and j 1 ̸ = j 2 , then the two segments are in the same wedge. In XZ projection, the two segments have the same origin and different targets, so they are interior disjoint in projection and also in 3D.

(iii) If i ′ 2 = i ′ 1 + 1, then the two segments are in two consecutive wedges issued from the same group. The upper segment of the first wedge is obtained for j 1 = 0 and has slope

z i ′ 1 -i 1 k i ′
1 -i1 (in the XZ projection plane). The lower segment of the second wedge is obtained for j 2 = k -1 and has slope

z i ′ 2 -i 2 k i ′ 2 -i2 = z i ′ 1 -i 1 +1 k-k+1 i ′ 1 -i1+1
. Denoting i = i ′ 1 -i 1 we ensure that the first wedge is below the second wedge by proving that

z i k i < z i+1 k -(k -1) i + 1 ,
which follows easily from (iv) If i ′ 2 > i ′ 1 + 1, the two segments are in different wedges issued from the same group. Their disjointness is granted by an easy induction from Case (iii).

z i+1 k -k+1 i+1 > z i+1 k -k i+1 = i+1 i z i + 1 -1 k i+1 ⩾ i+1 i z i k i+1 = z i k i . toto X Z v i ′ 1 ,⋆ v i2,⋆ v i1,⋆ e i1,⋆,i ′ 1 ,⋆ v i ′ 2 ,⋆ e i2,⋆,i ′ 2 ,⋆ Z z i k X v i ′ 1 j ′ 1 v i1j1
The disjointness of the downgoing parts of the edges (i.e., segments [e i1,j1,i ′

1 ,j ′ 1 ; v i ′ 1 ,j ′ 1 ] and [e i2,j2,i ′ 2 ,j ′ 2 ; v i ′ 2 ,j ′ 2 ]
) is obtained with similar arguments in Cases (i-ii) and, in Cases (iii-iv), because, since i

′ 2 > i ′ 1 , vertex v i ′ 2 ,⋆ is to the right of v i ′ 1 ,⋆
and bend point e i2,... is also to the right of bend point e i1,... (see Figure 5). The case of edges finishing within the same group of vertices is obtained by symmetry. Lemma 6. Refer to Figure 6. Two edges

E 1 = [v i1,j1 ; e i1,j1,i ′ 1 ,j ′ 1 ; v i ′ 1 ,j ′ 1 ] and E 2 = [v i2,j2 ; e i2,j2,i ′ 2 ,j ′ 2 ; v i ′ 2 ,j ′ 2 ]
properly intersects in XZ projection if and only if their endpoints are interlaced, that is,

i 1 < i 2 < i ′ 1 < i ′ 2 or i 2 < i 1 < i ′ 2 < i ′ 1 . If the bend or E 2 is above that of E 1 and i 1 < i 2 < i ′ 1 < i ′ 2 (resp. i 2 < i 1 < i ′ 2 < i ′ 1 )
, then the intersection occurs in the upgoing (resp. downgoing) part of E 2 .

Proof. We first show that, if i 1 ⩽ i 2 , the two edges properly intersects in XZ projection if and only if

i 1 < i 2 < i ′ 1 < i ′ 2 ; refer to Figure 6(left). Since i 1 ⩽ i 2 < i ′ 2 , integer i ′ 1 is in (i 1 , i 2 ], (i 2 , i ′ 2 ) or [i ′ 2 , ∞). In the first case, i 1 < i ′ 1 ⩽ i 2 < i ′ 2
, the edges are interior disjoint in X projection.

In the third case,

i 1 ⩽ i 2 < i ′ 2 ⩽ i ′ 1 , that is, both endpoints of E 2 are in between those of E 1 . If i 1 = i 2 or i ′ 1 = i ′ 2 the result follow from Lemma 5. Thus i 1 < i 2 < i ′ 2 < i ′ 1 . Since i ′ 1 -i 1 > i ′ 2 -i 2 , the bend of E 1 is above that of E 2 .
Thus, the left segment of E 1 is steepest than the left segment of E 2 (as in the proof of Lemma 5) and thus they do not intersect in XZ projection. Symmetrically, the downgoing edges of E 1 and E 2 do not intersect in XZ projection. It follows that the two edges do not intersect in XZ projection since their endpoints are nested.

Hence, the second case holds, that is

i 1 < i 2 < i ′ 1 < i ′ 2 .
The endpoints of E 1 and E 2 are thus interlaced and thus the edges intersect. The above argument on the slopes of the edge segments proves the second part of the lemma; see Figure 6(right).

Then, we have to determine the Y coordinate of the bends to avoid crossings. We do this algorithmically: for i, j, i ′ , j ′ from 0 to k. Using Remark 3, we choose the Ycoordinate y i,j,i ′ ,j ′ of the bend point on the edge from v i,j to v i ′ ,j ′ to be the smallest non-negative integer such that the edge properly intersects none of the edges that have been previously defined in the loop. Since the number of already drawn edges is O(n 2 ), y i,j,i ′ ,j ′ is granted to be also not bigger than O(n 2 ), even if we expect it to be much smaller. A similar incremental process is used by Wood [START_REF] Wood | Three dimensional graph drawing with fixed vertices and one bend per edge[END_REF] to determine the bends in a different context.

It should be noted that the bend points depend on the order in which we consider i, j, i ′ , j ′ . To optimize the resulting volume of the drawing, we define the bends in a lexicographic order, by first decreasing Z, then increasing X and increasing Y . In other words, in XZ projection (Figure 4), the bend points are defined from top to down and then from left to right. This is illustrated on Figure 6(right), downgoing wedges to consider against [v i1j1 e i1j1i ′

Experimental study

We implemented our construction to determine crossing-free 3D drawings of K n for n up to 10 000; precisely, we determined crossing-free 3D drawings of K k 2 for k ∈ [START_REF] Cohen | Three-dimensional graph drawing[END_REF]100]. The code is simple enough so that we are confident in its correctness; the code is given in Appendix D.

The values of Y n for n ⩽ 10000, given in Figure 7 and Appendix C, show a very reasonable increase of Y n in terms of n. Actually plotting Yn √ n log 2 n (see Figure 8) supports Conjecture 8.

We draw K [-25000, 90000] of volume about 6 times greater and also much worse aspect ratio. See Figure 9 for a more complete comparison of volume used by the two methods.

B Coplanarity anf XZ-leftturn predicates

We present here the computations for the predicates used in previous and next sections.

The usual determinants for 3D and 2D orientation simplify since some vertices are in the plane Z = 0. orient3D((x 0 , y 0 , 0), (x 1 , y 1 , 0), (x2, y 2 , z 2 ), (x 3 , y 3 , z 3 )) = x 0 x 1 x 3 y 0 y 1 y 3 0 0 z 3 1 1 1

= z 2

x 0 x 1 x 3 y 0 y 1 y 3 1 1 1

-z 3

x 0 x 1 x 2 y 0 y 1 y 2 1 1 1 = z 2

x 0 x 1 -x 0 x 3 -x 0 y 0 y 1 -y 0 y 3 -y 0 1 0 0 -z 3

x 0 x 1 -x 0 x 2 -x 0 y 0 y 1 -y 0 y 2 -y 0 1 0 0 = z 2 ((x 1 -x 0 )(y 3 -y 0 ) -(y 1 -y 0 )(x 3 -x 0 ))

-z 3 ((x 1 -x 0 )(y 2 -y 0 ) -(y 1 -y 0 )(x 2 -x 0 )).

The four points are coplanar if this determinant is 0.

orient2D((x 0 , 0), (x 1 , z 1 ), (x 2 , z 2 )) =

x 0 x 1 x 2 0 z 1 z 2 1 1 1 = x 0 (z 1 -z 2 ) + (x 1 z 2 -x 2 z1).

The three points turn left if this determinant is positive.

Figure 3 :

 3 Figure3: Drawing of the bi-partite graphs for 5 groups of 5 vertices in the Devillers-Everett-Lazard-Pentcheva-Wismath construction. Each edge has two segments, one from the vertex in the lower group to its bend in blue, another from the bend to the vertex in the upper group in red.

Figure 4 :

 4 Figure 4: New construction in projection in the XZ plane. Up-going wedges (in blue) are pairwise disjoint. Down-going wedges (in red or orange) are pairwise disjoint.

Figure 5 :

 5 Figure 5: For the proof of Lemma 5.

Figure 6 :

 6 Figure 6: Segment intersections in projection in the XZ plane.

1 j ′ 1 ]

 11 are in red, upgoing wedges to consider against [v i1j1 e i1j1i ′ 1 j ′ 1 ] are in purple, and upgoing wedges to consider against [e i1j1i ′1 j ′ 1 v i ′ 1j ′ 1 ]are in blue. The corresponding pseudo-code can be found in Appendix A.Theorem 7. A one-bend non-crossing 3D drawing of K n can be achieved within a box of size 2√ n × Y n × O(n log n), with Y n a function of n smaller than O(n 2 ).Proof. The distinct boxes that enclose the drawings of the complete graphs of each group are disjoint and below the plane Z = 0. The drawings of the bi-partite graphs between two groups are above the plane Z = 0 and thus bi-partite graphs and complete graphs do not interfere. Edges starting or finishing at the same place are pairwise disjoint by Lemma 5. Edges whose starting and finishing place are not interlaced are pairwise disjoint by Lemma 6. The remaining edges are pairwise disjoint by construction.The X-dimension of our construction is 2k = 2 √ n by construction. The Ydimension is some finite value Y n = O(n 2 ). The height is O(k 2 ) = O(n) below the plane Z = 0 by Lemma 1 and it is O(n log n) above the plane Z = 0 by Lemma 4. Conjecture 8. Y n = O( √ n log n) and thus the volume of the drawing in Theorem 7 is O(n 2 log 2 n) reaching the quadratic optimal bound up to log factors.

8 4 =Figure 7 :

 47 Figure 7: Y n for n ⩽ 10 4 .

Figure 8 :

 8 Figure 8: Yn √ n log 2 n for n ⩽ 10000.

Figure 9 :

 9 Figure 9: Volumes obtained in our construction (green) and Devillers et al. [5] (orange).

To deal with integer bounding boxes, we define the volume of[x, x ′ ] × [y, y ′ ] × [z, z ′ ]where all these bounds are integers to be (x ′ -x + 1)(y ′ -y + 1)(z ′ -z + 1) so that the volume is exactly the number of grid points in (and on the boundary of) the box.

A Pseudo code for determining the Y -coordinates of the bend points while (1){ N= K*K; // ensure N is a perfect square YMAX=K-1; printf("\nComplete graph with %d vertices\n",N); Z


Anna R Galliano
email: argaliano@univ-catholyon.fr

Serge Portalier
email: serge.portalier@univ-lyon2.fr

Annar Literaturereview

Galiano

Keywords: Cognition, Language, MentalRepresentations, VisualImpairment, Vision Langageetdéficiencevisuelle:unerevuedelittérature Résumé Cognition, Handicapvisuel, Langage, Représentationsmentales, Vision, directoroftheDepartmentofpsychologyCatholicUniversityofLyonUCLy

Thisarticleisareviewoftheliteratureontheeffectsofvisualimpairmentonlanguage.The article begins with an overview of the role of vision in language development from a philosophical point of view, followed by a psychological approach. The aim is to show that psychologyhasnotprovidedaclearanswertothequestionoftherelationshipbetweenvisual impairmentandlanguagedevelopmentinchildren. An inventory of specific linguistic features among blind children is done through a comparison with the language of blind adults. This inventory allows us to show that the specificfeaturesinchildrenarenotrealproblemsbutratheramodeofdevelopmentspecific toblindchildren.

shownthatvisionisnotalwaysessentialascanbeeasilyillustratedthroughtheexampleof blindness.Thisisnotanewfieldofthought,andthelinkbetweenknowledgeandvisionwas formulated by philosophers many centuries ago. The question has been examined from two viewpoints: those who advocate interdependence between vision and cognition and those whodenyit.

Over the last three centuries, philosophers declared that knowledge derives from experience. ThisthesiswassupportedbyBertrandRussell(1912)whoarguedthatthereisa kindofknowledgethatcannotbecodedintolanguage.Thisknowledgeissensoryand"direct" (knowledge by acquaintance), connected to the sensory system, and is the basis of the constructionofaphysicalworld.Withthistheory,ablindperson,deprivedofthisimportant component of experience (vision), apprehends the world in a different way from those who havenovisualimpairment.Consequently,theconceptualsystemmaybeconsidereddefective (Galiano&Portalier,2009).

TheantithesisissupportedbythephilosopherBryanMagee (1995),amongstothers,inan epistolary dialogue with the blind philosopher Martin Milligan. This thesis goes against the first and earlier theory, because it considers language as a mediator in the development of conceptual knowledge. This implies a "knowledge by description" which is constructed throughlanguage.Languagethusseemstohaveacompensationfunctioninthedevelopment oftheconceptualsystemofblindpeople.Therefore,theconceptualsystemofsuchindividuals isnodifferentfromthatofsightedpeople.

Thisbipolarapproachofthedebateaboutconstructionofconceptualknowledgeandthe roleplayedbylanguageinthisconstructionisalsopresentinpsychology (Galiano&Portalier, 2009).

APsychologicalView

Thedebateontheroleofperceptionandlanguageinthedevelopmentoftheconceptual systemofvisuallyimpairedpeopleisespeciallyinterestingincognitivepsychology,sincesuch aquestionrelatestosensoryimpairment.Justasinphilosophy,twoconflictingtheoriesexist in psychology: a theory of the primacy of perceptive experience over the linguistic one, and inversely,atheorythatadvocatestheprimacyoflanguageovervision. Psychologistshaverepeatedlystudiedtheroleoflanguageinthecognitivedevelopment of non-impaired individuals. In his first publications, [START_REF] Piaget | Sixétudesdepsychologie[END_REF] suggested that language plays a secondary role in the development of knowledge (thought precedes language), whereas Vygotsky (1985) advocates that language has a crucial place in cognitive development (language is seen as a tool for planning and controlling mental activity). Concerningblindpeople,twoschoolsofthoughthavealsodevelopedaroundthespecificissue ofsensoryimpairment (Galiano&Portalier,2009). [START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF]perspectiveconcurswithRussell'softhesupremacyofdirectexperience (knowledge by acquaintance). According to her theory, although blind people use the same wordsassightedindividuals,theexactsignificancegiventothesewordsdoesnotconsistently coincide with that of the sighted population. She explains this fact by the importance of perceptionintheformulationofconcepts;thesignificanceisdirectlylinkedtotheconceptual schemasofthespeaker. Therefore,avisualimpairmentleadstoanalteredunderstandingof anygivenword(resultingfromalackofrelationbetween'signified'and'signifier').

In opposition to this thesis is the idea of the primacy of linguistic experience over the perceptive experience. The latter theory does not deny the existence of a link between language and visual perception, but it does cast doubt upon the direct correspondence between cognition (or even knowledge) and language (Pérez-Péreira & Conti-Ramsden, 1999). Language is seen as a tool for understanding the world around us (knowledge by description).Todothat,blindpeoplehaveincreasedattentiontothespokenwordinorderto obtaininformationwithinthestructureofthesentencewhichtheycannotgaindirectlydueto their impairment. This theory is supported by [START_REF] Landau | Language and Experience: Evidence from the Blind Child[END_REF] and [START_REF] Gleitman | Thestructuralsourcesofverbmeaning[END_REF] with the syntactic bootstrapping hypothesis concerning sighted children. This hypothesis postulates that children are able to utilize their knowledge of the syntactic structureofasentencetoaccessthemeaningofwords.Hence,blindchildrenspeakthesame language as visually able individuals, because the information needed for grasping meaning doesnotusethevisualsystem (Landau&Gleitman,1985).However,theauthorsdospecify that for colour perception, blind children and sighted children do not fully share the same signification.Althoughblindpeopledousecolournames,thesignificationattributedtothese namesonlycoincidespartiallywiththoseofthevisuallyabled.Thelanguageofblindpeople specifically regarding colours is considered to be consensual in that they are able to use learned information about colours -for instance, when one asks blind people what colour grass is, they will answer "green" as they have been taught. However, the case of colour namingisanexceptionandgenerallythesyntacticcontextdoesprovideblindpeoplewiththe necessary information for the comprehension of words referring to colours (Landau & Gleitman,1985).Thereforeitseemsthat,dependingonthestancetakenonthisissue,authors giveanegativeorpositiveanswertothequestionoftheimportanceofasoundvisualsystem for the construction of conceptual knowledge. Concerning the system of conceptual developmentinblindpeople,researcherscontinuetodivergeastothesimilaritybetweenthe conceptualsystemoftheblindandthatofthesightedpopulation.

LanguageDevelopmentinBlindChildren

Researchinneurosciencehasshownthatverbalandtactilestimuliplayasignificantrole inthecompensationsystemofcongenitallyblindpeople (Burton,Snyder,Diamond&Raichle, 2002;[START_REF] Roder | Speechprocessingactivatesvisual cortexincongenitallyblindhumans[END_REF]Sadato, Pascual-Leone, Grafman, Ibanez, Deiber,Dold&Hallet,1996;Sadato,Pascual-Leone,Grafman,Deiber,Ibanez&Hallett,1998;[START_REF] Portalier | Développement de la personne déficiente visuelle[END_REF]. Language is an important source of information for the development of conceptual and analogical systems for the blind people. Previous studies about the developmentoflanguageinblindchildrenhaveprovidedawealthofinformation.Thisfocus onthelanguageskillsofblindchildrenenablesustogenerateaclearpictureoftheeffectsof visualimpairmentonlanguagedevelopment.

Several studies between the 1960s and the 1990s specifically explored the use of an egocentric style of language in blind children (Anderson, Dunlea & Kekelis, 1984;[START_REF] Urwin | Theacquisitionofcommunicativecompetence[END_REF]. An observation of their first hundred words suggests that blind children's speech is exclusively self-centred and largely composed of words describing the child's own actions. The language employed by these children does not therefore take external elements into account. Furthermore, during a conversation, blind children have a tendency to change or switch subjects [START_REF] Andersen | Blind children's language: Resolving some differences[END_REF]. A study of the effects of visual impairmentonfamilydiscussionsfocusedonfourmothersininteractionwiththeirblindor visuallyimpairedchildaged27-36monthsoverasix-monthperiod.Theresultsshowedthat the average length of speaking turns of the visually impaired children was similar to their mothers', while the average length of speaking turns of the blind children was considerably shorterthantheirmothers' (Kekelis&Prinz,1996).

A number of studies have aimed to illustrate one particular linguistic trait observed in blindchildren.Thesestudieshavefocusedonthedevelopmentalperiodbeforetheemergence of symbolic play, and sought to provide evidence of a specific language delay in the use of personal pronouns, and pronoun reversal problems [START_REF] Fraiberg | Parallel and divergent patterns in blind and sighted infants[END_REF][START_REF] Fraiberg | Self-represetation in language and play: Observations of blind children[END_REF][START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF]. [START_REF] Fraiberg | Aneducationalprogramforblindinfants[END_REF] found that whilst blind children developedthe"syncreticI"aroundthesameageastheirsightedpeers(aroundtheageof2.4 years), the "nonsyncretic I" appeared later, at around 4-5 years of age. For these experimenters, this difficulty in blind children is linked to the development of deixis, the exploitation of contextual information, in order to establish details about the individual's present, time and place. Developmental delays observed in the acquisition of the Theory of Mind in blind children could also be correlated with this difference in the acquisition of the pronoun "I", as well as the concept of "Me, not-Me" associated with the development of individualidentity,fromaveryyoungage.A"theoryofmind"isaspecificcognitiveabilityto understand others as intentional agents and to infer the beliefs, intentions, knowledge and desiresofothers (Baron-Cohen,1991).

Pérez-PereiraandConti-Ramsden(1999)suggestedthatthistendencytowardspronoun reversal [START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF][START_REF] Fraiberg | Parallel and divergent patterns in blind and sighted infants[END_REF][START_REF] Fraiberg | Self-represetation in language and play: Observations of blind children[END_REF] could not be said to be a specifictraitofblindchildren,asonlyasmallpercentageofblindchildrenregularlymakethis mistake. [START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF]showedthatsomevisuallyimpairedchildren,ataround30monthsofage, tendtospeakinthepasttense.Fromamorepragmaticviewpoint,concerningblindchildren's use of language in context, [START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF] longitudinal study of 6 children (two blind, two visuallyimpairedandtwosighted),revealedthatinblindchildren'sspeech,theproductionof statementstocapturetheattentionoftheotherpersonemergesbeforeoratthesametimeas affirmative statements. This tendency was not observed in the two sighted children. For Dunlea, this difference is due to the fact that sighted children already have visual cues and therefore less need to attract the other person's attention verbally. Another of Dunlea's findingsisthatunlikesightedchildren,blindchildrenarenotfoundtovocalisestatementsof offering and showing something. Similar results were observed in two other studies (McGinnis, 1981;[START_REF] Erin | Frequencies and types of questions in the language of visually impaired children[END_REF] highlighting an increase in the amount of questions blind children ask. [START_REF] Erin | Frequencies and types of questions in the language of visually impaired children[END_REF] for example, using a similar methodology with three groups of children aged between four and ten, found that although all three groups showed a high incidenceofrequests,(wordsbeginningin"Wh")thepercentagewasslightlyhigherinblind orvisuallyimpairedchildren.Accordingtohim,thisisduetothefactthat"Wh-"questionsare essentialandaretheonlywayfortheblindchildrentodiscovertheirsurroundings. Anderson & Ficher (1986) carried out a study of 10 congenitally blind children and 10 childrenwithnovisualdisability,allagedthreetofour.Thestudyfocusedonnominalrealism, thetendencytoconsiderthenamegiventoanobjectandtheobjectitselfasoneandthesame. Theyobservedthatnominalrealismaswellasanimismpersistmuchlongerinblindchildren than in sighted children. As for spoken language, which is an indication of a good understanding, [START_REF] Mcconachie | Early Language Development and Severe Visual Impairment[END_REF] observed a delay in blind children due to a delay in acquisition of the first words around the age of two. A 1992 longitudinal study by Pérez-Pereira and Castro, followed two dizygotic twin Hispanic girls, one blind and one sighted, betweentheagesof2.5and3.5.Thisstudyillustratesthedifferencesintheuseofpragmatic categories. Specifically, the blind child tended to use more self-centred language with very little use of language directed toward others. Furthermore, she uttered significantly more statements describing actions or her intention to carry out actions or wishes. Results also showed that the blind sister often used language imitation, routine and repetition (IRR speech). These findings were complemented by a subsequent publication concerning the same two children (Perez-Pereira and Castro, 1997) (a blind child and her sighted twin sister),thistimeupuntiltheageof5.4years.Thefindingsofthissecondstudyconfirmedthe importance of imitation, routine and repetition in the blind child. The authors state that children appear to use a "Gestalt style of language learning", as a mechanism for language analysis,aparticularlyvaluabletechniqueforblindchildren. Someauthorshavedefinedthelinguisticperformanceofblindpeopleasincongruousdue to a lack of visual cues. It is helpful to review the debate of the use of colour words. The concept of verbalism was first introduced by Cutsforth in 1932 (in [START_REF] Hatwell | Psychologiecognitivedelacécitéprécoce[END_REF]Pérez-Péreira & Conti-Ramsden, 1999) to describe words or expressions used by blind people withoutreferencetoaconcretesensoryexperience.Blindpeoplecanthusemploythewords ofthesightedpeopletodescribeobjectsthattheyhavenotbeenabletoexperiencedirectly due to their visual impairment. Verbalism has since been separated into visually oriented verbalism, for words arising from a visual experience, and global verbalism covering those words that are not associated with a sensory experience [START_REF] Harley | Verbalismamongblindchildren:Aninvestigationandanalysis[END_REF]. Harley (1963) compared40childrenagedbetween6.11and14.3onaverbalismscale(worddefinition).The resultsofthisstudyindicatedthatverbalismdecreaseswithage. Demott's(1972)studyof41 blindpeople,41visuallyimpairedindividualsand61sightedparticipantsagedbetween6and 19years,wasalsobasedonaverbalismscale.Hisresultssuggestthatthereisnosignificant difference between groups. The sighted individuals generally gave the same meaning to concepts,justastheblindandvisuallyimpairedpeopledid.

However, several studies [START_REF] Anderson | Word meaning among blind children[END_REF][START_REF] Millar | Language and active touch: some aspects of reading and writing by blind children[END_REF][START_REF] Nelson | Making sense: The acquisition of shared meaning[END_REF][START_REF] Nolan | Ontheunrealityofwordstotheblind[END_REF] raised doubts on the existence of such a specific language trait in the blind, as even sightedpeoplemakeuseofwordsofwhichtheyhavenodirectconcreteexperience.In2005, Rosel,Caballer,JaraandOliverpublishedastudytoinvestigatetheuseofsuchverbalismin 62congenitallyblindchildrenand64visuallyimpairedchildrenagedbetween7and14.They foundthatneitherthedegreeofvisualimpairmentnorthechild'sgenderhadanyimpacton thefrequencyofverbalismuseinthechildren.Age,ontheotherhand,wasfoundtohavea significant impact. Such results suggest that language is a flexible structure correctly employedbycongenitallyblindchildren;thelatterhavetheabilitytoadaptadequatelytothe languageusedbythosearoundthem.

Many studies on language acquisition by blind children (see [START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF][START_REF] Fraiberg | Self-represetation in language and play: Observations of blind children[END_REF] highlight delays rather than true disorders. According to these authors, although the visually impaired children do not necessarily develop linguistic abilities at the samerateasthesightedchildren,theyacquirethemeventually.Otherauthors (Pérez-Pereira &Castro,1997;Pérez-Pereira&Conti-Ramsden,1999)supporttheideaofspecificlinguistic traits in the blind. They insist on the importance of taking into account the linguistic differencesobservedasevidenceofanalternativeandspecificpatternofdevelopmentrather thanadeficit.

LanguageUseinBlindAdults

Theissueoflanguageuseinblindadultshasseeminglyattractedlittleresearchinterest. Studies in this area are scarce and far between and are mainly carried out in social psychology, most notably the work of [START_REF] Rutter | Visual communication and the content andstyleofconversation[END_REF], [START_REF] Kemp | Social psychological aspects of blindness: A review[END_REF], [START_REF] Rutter | Lookingandseeing:TheRoleofVisualcommunicationinsocialinteraction[END_REF], and [START_REF] Kemp | SocialInteractioninBlindPeople:AnExperimentalAnalysis[END_REF]. Furthermore, these studies do not aim to understandthespecificnatureoflanguageinblindpeople,butratherstudytheroleofvision incommunication.Theuseofagroupofblindparticipantsenablestheauthorstounderstand the role played by vision in conversational interactions as well as observe the behaviour of blindpeopleinsocialinteractions.Theseauthorshavesuggestedtheconceptofcuelessnessin reference to the ensemble of multisensory non-verbal cues available and exploitable by the speakersintheconversation (Rutter,StephensonandDewey,1981;Rutter,1984).Therefore, itispossibletoratesocialinteractionsonanordinalscale,baseduponthequantityofmultisensorynon-verbalcuesavailabletoeachspeaker.

In1981Rutter,StephensonandDeweyusedtwoexperimentalconditions:aface-to-face interactionversusatelephoneinteraction.Thesecondsituationintroducesthevariableofthe absence of reciprocal visibility that can be linked to blindness. The authors analysed some thirtyvariables,andtheresultsshowedveryfewdifferencesbetweenthetwosituations.

Another study published by [START_REF] Kemp | Social psychological aspects of blindness: A review[END_REF]Rutter (1986) (in Kemp 1981;[START_REF] Rutter | Lookingandseeing:TheRoleofVisualcommunicationinsocialinteraction[END_REF]) included blind people using three experimental conditions in the following way: dyadic interactionbetweentwoblindpeople,betweenoneblindpeopleandonesightedperson,and finally, between two sighted people. The three situations were used to observe the effect of non-verbalcuesonspeechproduction.Theconditioninwhichtwoblindindividualsinteract containstheleastnon-verbalcuestosupportthecommunication.Conversely,theinteraction between the two sighted participants involved the most non-verbal cues, the blind-sighted interactionbeinghalfwaybetweenthetwo.

The results of this study show that the statements spoken by the blind group are less instrumental and more personalised when compared to the sighted dyads. In regard to language style, blind people were more spontaneous and hence less structured than the sightedpeople.Significantdifferenceswereobservedinthefollowingvariables:'offeringand accepting information', 'procedure and irrelevant information', 'duration of simultaneous speech(seconds)'and'numberofutterances(words)'.Onaverage,theexperimentalgroupof blind-blind dyads obtained higher rates on most of these variables. However, the variable 'offering information' was found to be inferior in this group compared to the other two groups.Resultsalsoshowedthat,statistically,thegroupofblind-sighteddyadswashalfway between the other groups. Contrary to expectations, the blind-blind group actually showed very small differences compared to the other groups (blind-sighted and sighted-sighted) in theirspeechbehaviours.

In1998MacCluskie,Tunick,Dial&Paulstudiedtheroleofvisioninthedevelopmentof abstraction, the ability to formulate categories and concepts and express them through language.Theyparticularlylookedatdifferencesinverbalandnon-verbalcommunicationin the abstractive ability of adults who were blind from an early age (before the age of two) comparedtothosewholosttheirsightaftertheageoffive,whentheirexpressivelanguage skills were more developed. No significant differences were observed between the two groups,withregardtotheseelementsassessedusingtheCognitiveTestfortheBlind(CTB) and the Verbal subscale of the Wechsler Adult Intelligence Scale-Revised (WAIS-R). Furthermore, researchers observed that the level of education affected the variability of scoresinWAIS-Rtests.

Discussion

Aswehaveobservedinthisarticle,researchersdonotagreeonthenatureoftheeffectsof visual impairment on language, and on the part played by language in the construction of meaning.However,wecanassertthatvisualimpairmentdoesnotnecessarilyinvolvespecific languagedisorders.Surveyswehavelistedarefewbutrevealthatvisualimpairmentinvolves difficulty in acquiring language elements, rather than true language disorders. Moreover, difficulties arising during childhood often either subside or even totally disappear in adulthood.Nevertheless,thosesurveysalsohavetheirlimitations.First,theirsamplesizesare too small (only a few subjects per study), and they should rather be considered as case studies.

Their results cannot be generalised and applied to all blind people. Secondly, the comparisonparameterusedinthesesurveysisapopulationofsightedpeople.Eventhough this choice enables the differences between both populations to be highlighted and the deficitstoberecordedforexample,thesesurveysdonotindicatewhatleveloflanguageany specific blind subject has reached within the visually impaired population. What we need indeedisanexplanatorymodelofthetypicallanguagedevelopmentinblindsubjects,which we could refer to when making a diagnosis. The necessity for carrying out research in this fieldseemsobvious.Morespecifically,wesuggestthatresearchshouldaddresstwospecific languageaspects.

First, we know that blindness involves difficulties of spatial representation (see Hatwel, 2006). It would be relevant to check if this characteristic exists in the acquisition of spatial language.In2009,theINSHEAinSuresne(France),underthesupervisionofthelinguistN. Lewi-Dumont, initiated a research project on spatial language acquisition at school, from kindergartentoelementaryschool.Thefirstresultsareexpectedattheendof2011.Wethink that this type of research should be extended to other linguistic areas in order to study the influenceofalinguisticcomplexityfactor.

The second field of research is pragmatismthat is the use of contextual language -, whichrepresentsthethirddimensionoflanguage,aftermeaningandsyntax.

Moststudiesonlanguagefocusonthesetwolatterdimensions.Veryfewstudies [START_REF] Dunlea | Vision and the Emergence of Meaning: Blind and sighted children's early language[END_REF]Pérez-Péreira & Castro, 1992) address the pragmatic dimension of language. Studies should focus more specifically on the referential function of language [START_REF] Galiano | Lesfonctionsdulangagechezlapersonneaveugle.Metaanalysedelarelationentreconnaissanceetlangage[END_REF],thatistosaythewayablindpersondescribessensiblethings.Developingresearchon the referential function of language would lead to an improvement in audio description technology (narrative description of films, documentaries, plays, etc, for blind and visually impaired audiences). In fact, such a technology is based essentially on a few ministry recommendationsandtheempiricalpracticeoftranslators (Galiano&Portalier,2010).

Finally, initiating research in these two areas would help increase our knowledge of languagedevelopmentspecifictoblindpeopleandthusimprovetheireverydaylife.